LOVE GOD, LOVE THOSE HE LOVES

LOVE GOD, LOVE THOSE HE LOVES

(A FRIEND STORY)

The expression “Love me, love my dog” came to life when my wife and I got a puppy—a first for me. We adore Sophie and she adores us. Actually, she adores everyone. She is not only one of those hyperactive toy breeds, but also one that is famously sociable. She starts wagging her tail as soon as a new person enters her world, and within a second or two her tail is wagging the rest of her so hard that she nearly comes unglued. If the new person so much as acknowledges Sophie’s existence, she gets even more excited and eager to “bond.” This is when we find out who loves dogs and who doesn’t.

I’m about to make a larger point, but before I do, another little story that I think most parents will be able to relate to. When my children were small they acted, well, childish. They whined and cried over the littlest things, spilt more food and drink than they swallowed, broke stuff out of clumsiness and curiosity, and as soon as they were old enough to understand boundaries, began pushing them. I loved them anyway. They could be annoying, even aggravating, but they were only children, after all, and this was all part of the learning, maturing process. Plus they were mine. What really annoyed me was when others let their annoyance show. “Love me, love my children.”

In that context, the connection between the two rules for life that Jesus said encompass all the rest—love God, and love others1—becomes clearer. “Love God, love those He loves,” which is everyone. If we truly love God and believe that He created us in His own image, as the Bible says, we will love and respect each of His creations enough to try our best to understand and accept him or her—faults, foibles, and all.

—

Matthew 7:12 ESV / “So whatever you wish that others would do to you, do also to them, for this is the Law and the Prophets.

Romans 12:10 ESV / Love one another with brotherly affection. Outdo one another in showing honor.

Philippians 2:3 ESV / Do nothing from rivalry or conceit, but in humility count others more significant than yourselves.

Onyedikachi Kingsley Ogbonna

How to Have a Quiet Time as a Couple

How to Have a Quiet Time as a Couple

Phylicia Masonheimer

I’ll be honest: Josh and I have probably tried eighteen different “quiet time” formats in the last two years. It’s hard enough finding time for your own devotions; add another person’s schedule and the variables are endless!

Despite the difficulty, we continue to make an effort in this area. Whether you’re dating or newly married, pursuing God together is the surest way to build a strong relationship foundation. If you’re dating, spending time in God’s Word as a couple will strengthen your resolve for purity. If you’re married, regular devotional times will transform your attitudes toward one another. It’s no wonder the Enemy targets couples with distraction and busyness every time they sit down to pray!

Josh and I haven’t “arrived” in this area, but we’ve found some things that work. Despite three years of inconsistent work schedules, moving three times, and balancing work, home, and a baby, we’re proof that a quiet time is possible if you’re intentional about it. The following four principles are very simple, but it’s the simplicity that makes them work.

DESIGNATE A TIME

When we were dating, our “quiet time” was usually before each date. We found that reading and praying together strengthened the Spirit’s voice in our hearts when we were tempted to compromise. Every time we struggled with purity, we hadn’t spent time seeking God together. We hadn’t sought His protection over our hearts and minds. Even if we were getting together to watch a movie or eat out, we tried to make the first thing on our agenda a prayer time.

Once married, quiet times actually became harder. We worked two different shifts, we both traveled on business, and there were many weeks where the only time we saw each other was at bedtime. We realized it was more important to have the devotional time than to have it at the same time each day. Rather than saying, “We’ll pray together at 6 AM each morning,” we decided day-to-day what time worked best.

In some seasons, we read and prayed at night, when we were for sure together. In others we’d get up together and read over breakfast (which is what we do now). Find a time that works for you both.

It’s worthy to note that it is not “usurping leadership” to remind your boyfriend/husband of your commitment to meet God. That accountability will be very necessary. If he continues to be negligent about pursuing the Lord, 1) if you’re dating – ask yourself if you are actually equally yoked; 2) if you’re married, pray that God would transform his heart and give him a spirit of leadership. I have seen this prayer answered many times over when Josh was not leading as much as I’d wish. Rather than nagging, ask God to motivate him to step up.

PICK A PASSAGE

Josh and I have used a few books and devotionals to guide our quiet times, but we prefer to use the Bible itself. While devotionals provide some structure, they don’t get you in the Word of God – the seat of God’s power and influence.

Right now, Josh and I are doing my New Testament reading plan each morning. We take turns reading a passage using different versions of the Bible. This gives us a different perspective and promotes conversation.

If you’re just starting out, pick a small book (like one of the NT epistles) and read it in chunks. Don’t rush; you don’t need an end date. If you want to discuss verse by verse, do so! What matters is that you are in the Word together.

ASK QUESTIONS

My favorite Bible study method is SOAP: Scripture, Observation, Application, and Prayer. If you’re a couple who needs structure, this can be applied to any passage you read. It can also give you a groundwork for questions.

There is no such thing as a dumb question about the Bible. As you discuss the passage, any questions you can’t answer should be written down for future research (or better yet, research them together during your quiet time!). This doesn’t have to take long. Two or three questions will get you thinking deeper about the passage.

It’s important to engage with the text and with each other. Our temptation is to simply read the words and check “quiet time” off the list, but that’s not the purpose of it. For God to change us through His word, we need to “camp out” in it. This doesn’t mean you’re having an hour devotional time. Just make sure the time you have is completely focused on what you’re reading.

ENGAGE YOUR CHILDREN

Even if you don’t have kids yet, consider this habit of a couple’s quiet time as an investment in your future family. If you’re unable to make it a habit as a couple, it will be ten times harder with kids!

Now that Josh and I are parents, devotions take more effort. But we try to model this practice in Adeline’s view, and even include her in our study. At thirteen months old she may seem too young for what we’re doing, but the mere act of involving her creates an environment of discipleship.

One of my favorite resources for beginning discipleship is Addie’s Say and Pray Devotions by Diane Stortz. This book provides structure to a family “quiet time” while being age appropriate for babies. Each page contains a one sentence “devotion” and a short Bible verse. The illustration is pertinent to the devotional subject, and each object on the page is labeled so the child can learn new words. Adeline loves this book and I love reading it to her. It is a creative way to get her accustomed to God’s Word without going too far above her head.

Head over to my Instagram to enter a giveaway to win a Say and Pray Devotional for your baby or one you know!

ALWAYS CONCLUDE WITH PRAYER

Finally, always conclude your quiet time with prayer. Josh and I actually break this up; we read in the morning and pray together at night. Since Josh leaves for work right after we read together, we found it was better to pray before going to sleep. We’re guaranteed to be in the same place, and we can reflect on the passage we read throughout the day.

I know praying together sounds trite. Of course you should pray together – you’re a Christian couple! But honestly, how many of us truly make time to do this? Prayer is not so much about us as it is about God: Exposing our hearts to His transforming power. You cannot speak to God and remain unchanged. That’s why every Christian relationship needs prayer.

Don’t be ashamed of small beginnings in this area. Like I said: Josh and I are works in progress in this area! You will never get to a devotional plateau, with no more to learn about God or the Bible. What matters is that you try.

How to Grow Strong in Your Faith

How to Grow Strong in Your Faith

by Mark Altrogge

In Romans 4, Paul tells us Abraham “grew strong in his faith” and urges us to walk in Abraham’s footsteps. To believe like he believed. How do we do this?

In hope he believed against hope, that he should become the father of many nations, as he had been told, “So shall your offspring be.” He did not weaken in faith when he considered his own body, which was as good as dead (since he was about a hundred years old), or when he considered the barrenness of Sarah’s womb. No unbelief made him waver concerning the promise of God, but he grew strong in his faith as he gave glory to God, fully convinced that God was able to do what he had promised. (Romans 4:18–21)

Look to God’s promise not your circumstances.

In hope he believed against hope, that he should become the father of many nations (18)

Abraham’s situation looked bleak. God promised him multitudes of descendants, but the only problem was he was well past child producing. “He considered his own body, which was as good as dead (since he was about a hundred years old).” He also considered the barrenness of Sarah’s womb. Not only was Abraham almost 100 years old, but Sarah his wife was very old, and she had never been able to have children her whole life. How are they going to have children? If Abraham had based his hope on his circumstances he would have given up. But In hope he believed against hope—God’s promise gave him hope in his hopeless situation. He put his hope in God’s promise, not his circumstances.

We may feel hopelessly unrighteous. We may feel like God could never forgive us for the sins we have committed, that he would never accept us. But we must not look at ourselves, just like Abraham didn’t look at himself, but like Abraham, we must believe God’s promise of grace. He counts me righteous in Christ!

Our teenager may seem hopelessly lost. Our finances may be out of control. We may lack direction for our lives. Our marriage might be frustrating or our church might be a mess. Look to Jesus Christ! Don’t look to yourself. Look to the promise of the gospel—everyone who believes in him shall be saved. Look to his promises to draw near to those who draw near to him. Promises to hear and answer our prayers.

Give glory to God

No unbelief made him waver concerning the promise of God, but he grew strong in his faith as he gave glory to God, fully convinced that God was able to do what he had promised. (20–21)

Abraham strengthened his faith. Here’s how: “He grew strong in his faith as he gave glory to God.” Begin to give glory to God—start thanking and praising him for his every promise. Thank him for saving you and declaring you righteous in him. He has promised to be with us when we pass through the waters and walk through fire. He has promised to never leave us nor forsake us. He has promised that nothing will be able to separate us from his love. He has promised to give us everything we truly need to glorify him. He has promised that we can do all things he requires through Christ who loves us. Praise him for these things!

We can look to our circumstances—it may not LOOK like God is being faithful. It may not FEEL like God is with us in these waters. It may FEEL like he has abandoned or forsaken us. We may not SENSE his love. But WE MUST NOT WAVER CONCERNING THE PROMISE OF GOD! Rather, we grow strong in our faith as we GIVE GLORY TO GOD, as we are fully convinced that God is able to do what he had promised.

In Ps 43 the Psalmist says “Why are you cast down O my soul? Hope in God for I shall yet praise him.” Keep thanking God, keep praising him in faith in the midst of your hard times. Say, “Jesus thank you that you are with me. Thank you have promised that your steadfast love never ceases. Praise you that your mercies never come to an end; they are new every morning. Great is your faithfulness.”

Growing stronger in our faith is not complicated. Look to God’s promise and glorify him. So, what are you going to believe today—God’s word or your circumstances? God’s promises or your feelings? God’s bedrock pledge of faithfulness or your wavering emotions? Walk in the footsteps of Abraham and strengthen your faith.

Lesson Note on methods used to determine azimuth by observations of the sun.

At what time will the effects of a small error in the determination of observer's latitude be minimized when making azimuth observations on Polaris?
A. 12:00 midnight
B. when Polaris is at elongation
C. when Polaris is at culmination
D. when the LHA is 90 degrees
E. when the GHA is 90 degrees
Ans; C. At culmination the bearing is 0 at all LAT.

There are two methods by which azimuth can be determined by observation of the sun. Answer the following questions concerning these methods.


QS > Name the two methods that can be used to determine azimuth by observations of the sun.
Ana; 1. Altitude method and
           2. hour angle method.

QS>Which method is more accurate? Explain your answer.
Ans; Hour angle method—because time can be very accurately determined, and inaccuracies in measuring the vertical angle and determining refraction make the altitude method less reliable.

Is Grace a License to Sin?

Is Grace a License to Sin?
The question “Is Grace a License to
Sin?” has presented itself in the form of
numerous questions and statements in the
past, such as the following:
Does preaching Grace lead to a
careless lifestyle?
Won’t people run out and sin
recklessly if they get too “deep”
into the Grace message?
We can not preach Grace to a
new believer because they are
not mature enough to be trusted
with such freedom yet.
Taking the Law off believers gives
them a license to sin.
All these statements fly straight in the
face of what the Bible teaches about
Grace:
Tit 2:11 For the grace of God that brings
salvation has appeared to all men, 12
teaching us that, denying ungodliness and
worldly lusts, we should live soberly,
righteously, and godly in the present age.
(KJV)
Firstly let us consider how Grace
influences a person’s thinking. When we
comprehend what the message of Grace is
all about (how God pardoned the sins of
the whole world (Hebrews 10:17),
abolished the written code of the Law
(Colossians 2:14), set people at liberty to
live free from the fear of judgement and
punishment (1 John 4:18), how believers
are encouraged to have boldness when
approaching God (Hebrews 10:19-21), that
we can be confident that God will never
be angry with us ever again (Isaiah
54:9-10) and many other truths like these),
it is clear that a proper understanding of
these issues will in fact not encourage a
person to want to sin, but rather inspire
such a person to be more thankful towards
God for all He has done, to live a life
worthy of the sacrifice made by Jesus and
to deny ungodly conduct as stated in Titus
2:11-12 (above).
Mostly the arguments and questions
against the Grace message, such as the
ones we mentioned above, come from
people who are not necessarily afraid that
they themselves will be deceived into
wanting to commit more sins, but their
arguments are for other “weaker”
Christians who in their opinion do not yet
have the “maturity” to handle the
responsibility that comes with such
freedom. Therefore they encourage that
the truth of the Grace message be taught
with a healthy dose of Law mixed in to
warn these “weak” Christians against the
perils of sinning. How ironic isn’t it then
that the Bible teaches us that sin doesn’t
increase through Grace, but rather through
the Law:
Rom 5:20 God’s law was given so that all
people could see how sinful they were. But
as people sinned more and more, God’s
wonderful grace became more abundant
(NLT).
Grace actually came in and covered the
sins that were made more abundant
through the Law!
So for someone to say that Grace is a
license to sin, it simply serves to expose
such a person’s ignorance to what Grace
actually means and stands for. It reveals
that they don’t understand that the power
of the Holy Spirit inside a believer
(reminding them of their complete 100%
righteous standing before God) is an
infinitely stronger empowerment for “good
behavior” than threatening someone with
the Law. As a result the legalists can all
relax and come to terms with the fact that
the Holy Spirit can be trusted with the
transforming work in the life of a believer:
Phil 1:6 being confident of this very thing,
that He who has begun a good work in you
will complete it until the day of Jesus
Christ. (KJV)
Nobody appointed us as each other’s
moral policemen, so how can we ever try
to assume that function?
A good response whenever we hear
someone ask “Is Grace a licence to sin?”,
is whether a fear of punishment is their
only motivation for doing good. We do
good works and live a moral lifestyle
because we WANT TO (the desires of the
Holy Spirit working in and through us), not
because we fear the consequences if we
don’t.
The truth is that Grace can never be
reduced to an “acceptable” level to
compensate for the insecurities of
legalists. Due to the extreme nature of the
Law, Grace needs to be preached in its
purest form as well in order to free people
from the poison of “works based”
mindsets.
Check out this awesome article by Bas
Rijksen: Should you be careful not abuse
Grace? NEVER.

Question: "What is the baptism of/by/with fire?"

Question: "What is the baptism of/by/with fire?"

Answer: John the Baptist came preaching
repentance and baptizing in the wilderness of
Judea, and he was sent as a herald to announce
the arrival of Jesus, the Son of God ( Matthew
3:1-12 ). He announced, “I indeed baptize you
with water unto repentance, but He who is
coming after me is mightier than I, whose
sandals I am not worthy to carry. He will
baptize you with the Holy Spirit and
fire” ( Matthew 3:11 ).
After Jesus had risen from the dead, He
instructed His apostles to “…wait for the
Promise of the Father which you have heard
from Me; for John truly baptized with water, but
you shall be baptized with the Holy Spirit not
many days from now” ( Acts 1:4-5 ). This promise
was first fulfilled on the day of Pentecost ( Acts
2:1-4 ), and the baptism of the Spirit joins every
believer to the body of Christ ( 1 Corinthians
12:13 ). But what about the baptism with fire?
Some interpret the baptism of fire as referring
to the day of Pentecost, when the Holy Spirit
was sent from heaven. “And suddenly there
came a sound from heaven, as of a rushing
mighty wind, and it filled the whole house where
they were sitting. Then there appeared to them
divided tongues, as of fire, and one sat upon
each of them” ( Acts 2:2-3 ). It is important to
note that these were tongues as of fire, not
literal fire.
Some believe that the baptism with fire refers to
the Holy Spirit’s office as the energizer of the
believer’s service, and the purifier of evil within,
because of the exhortation “Do not quench the
Spirit” found in 1Thessalonians 5:19 . The
command to the believer is to not put out the
Spirit’s fire by suppressing His ministry.
A third and more likely interpretation is that the
baptism of fire refers to judgment. In all four
Gospel passages mentioned above, Mark and
John speak of the baptism of the Holy Spirit,
but only Matthew and Luke mention the baptism
with fire. The immediate context of Matthew
and Luke is judgment ( Matthew 3:7-12 ; Luke 3:
7-17 ). The context of Mark and John is not
( Mark 1:1-8 ; John 1:29-34 ). We know that the
Lord Jesus is coming in flaming fire to judge
those who do not know God ( 2 Thessalonians
1:3-10 ; John 5:21-23 ; Revelation 20:11-15 ), but
praise be to God that He will save all that will
come and put their trust in Him ( John 3:16 )!

Question: "Should a new believer be baptise immediately?

Question: "Should a new believer be baptise immediately?"
Answer: In the New Testament, new Christians
were often baptized immediately after
confessing Jesus as Lord. Should churches
continue this practice today? Two issues need
addressed. First, can new believers be baptized
immediately? The biblical answer is a definite
yes.
Three thousand believers were baptized on the
same day they believed when the church began
at Pentecost ( Acts 2:41 ). The Ethiopian with
Philip was baptized the same day he believed
( Acts 8:26–38 ). Paul (then Saul) was baptized
about three days after experiencing Jesus on
the road to Damascus ( Acts 9). Acts 16:15
shows a woman baptized the same day she
believed. Acts 16:33 notes the Philippian jailer
and his family were baptized the night they
believed. The first 3,000 people added to the
church were baptized ( Acts 2:41 ), and Jesus
commanded His followers to baptize other
disciples ( Matthew 28:19 ). Baptism is clearly
something expected of every Christian, whether
or not they are baptized immediately.
The second issue to address, however, is
whether a new believer is required to be
baptized immediately. Some churches argue
against spontaneous baptisms due to past
examples of people being baptized without a
true understanding of the meaning of salvation.
To prevent confusion, these churches offer a
class or other instructional time to help each
person understand these issues prior to
baptism.
Historically, during the third and fourth centuries
the theology of baptism continued to shift in
church practice. Originally, church instruction
took place after baptism. However, as different
heresies started to confront the church,
believers were increasingly given specific
instructions before being baptized. By the fourth
and fifth centuries, several weeks were required
to teach catechism before baptism. Because no
direct command is given in Scripture regarding
the length of time required between a person’s
confession of faith and his baptism, there is
freedom for each church and its leaders to
develop the best practice for their particular
congregation.
Though there is no requirement regarding
immediate baptism, there seems to be a clear
emphasis on closely associating a person’s
confession of faith and baptism. Therefore, a
church would do well to keep the space of time
between a person’s confession of faith and
baptism as short as possible. Further, many
churches do not allow a person to partake in
communion, become an official church member ,
or other important aspects of church life until
after baptism. These factors further add to the
importance of holding baptisms for new
believers in a timely manner.

Question: "What does the Bible say about being a good parent?"

Question: "What does the Bible say about being a good parent?"

Answer: Being a good parent can be a difficult and challenging venture, but at the same time can be the most rewarding and fulfilling thing we ever do. The Bible has a great deal to say about the way we can successfully raise our children to be men and women of God. As a good parent, the first thing we must do is teach them the truth about God's Word.

Along with loving God and being a godly example by committing ourselves to His commands, we need to heed the command of Deuteronomy 6:7-9 regarding teaching our children to do the same. This passage emphasizes the ongoing nature of such instruction. It should be done at all times—at home, on the road, at night, and in the morning. Biblical truth should be the foundation of our homes. By following the principles of these commands, we teach our children that worshiping God should be constant, not reserved for Sunday mornings or nightly prayers.

Although our children learn a great deal through direct teaching, they learn much more by watching us. This is why we must be careful in everything we do. We must first acknowledge our God-given roles. Husbands and wives are to be mutually respectful and submissive to each other (Ephesians 5:21). At the same time, God has established a line of authority to keep order. “Now I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God” (1 Corinthians 11:3). We know that Christ is not inferior to God, just as a wife is not inferior to her husband. God recognizes, however, that without submission to authority, there is no order. The husband's responsibility as the head of the household is to love his wife as he loves his own body, in the same sacrificial way that Christ loved the church (Ephesians 5:25-29).

In response to this loving leadership, it is not difficult for the wife to submit to her husband's authority (Ephesians 5:24; Colossians 3:18). Her primary responsibility is to love and respect her husband, live in wisdom and purity, and take care of the home (Titus 2:4-5). Women are naturally more nurturing than men because they were designed to be the primary caretakers of their children.

Discipline and instruction are integral parts of being a good parent. Proverbs 13:24 says, “He who spares the rod hates his son, but he who loves him is careful to discipline him.” Children who grow up in undisciplined households feel unwanted and unworthy. They lack direction and self-control, and as they get older they rebel and have little or no respect for any kind of authority, including God's. “Discipline your son, for in that there is hope; do not be a willing party to his death” (Proverbs 19:18). At the same time, discipline must be balanced with love, or children may grow up resentful, discouraged, and rebellious (Colossians 3:21). God recognizes that discipline is painful when it is happening (Hebrews 12:11), but if followed by loving instruction, it is remarkably beneficial to the child. “Fathers, do not exasperate your children; instead, bring them up in the training and instruction of the Lord” (Ephesians 6:4).

As a good parent, it is important to involve your children in the church family and ministry when they are young. Regularly attend a Bible-believing church (Hebrews 10:25), allow them to see you studying the Word, and also study it with them. Discuss with them the world around them as they see it, and teach them about the glory of God through everyday life. “Train a child in the way he should go, and when he is old he will not turn from it” (Proverbs 22:6). Being a good parent is all about raising children who will follow your example in obeying and worshipping the Lord.

Lesson Note On Solving plane triangles

Solving plane trianglesEdit

Standard notation for a triangle

A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:

• Three sides (SSS)
• Two sides and the included angle (SAS)
• Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length.
• A side and the two angles adjacent to it (ASA)
• A side, the angle opposite to it and an angle adjacent to it (AAS).
• Three angles (AAA) on the sphere (but not in the plane).

For all cases in the plane, at least one of the side lengths must be specified.

 If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution

Trigonomic relations

The standard method of solving the problem is to use fundamental relations.

Law of cosines

${\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha }$

${\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta }$

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }$

Law of sines

${\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}$

Sum of angles

${\displaystyle \alpha +\beta +\gamma =180^{\circ }}$

Law of tangents

${\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}$

There are other (sometimes practically useful) universal relations:

 the law of cotangents and Mollweide's formula.

NotesEdit

1. To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1).
2. We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.

Three sides given (SSS)Edit

Three sides given

Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:

${\displaystyle {\begin{aligned}\alpha &=\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}\\[4pt]\beta &=\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.\end{aligned}}}$

Then angle γ = 180° − α − β.

Some sources recommend to find angle β from the law of sines but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one.

Another method of calculating the angles from known sides is to apply the law of cotangents.

Two sides and the included angle given (SAS)Edit

Two sides and the included angle given

Here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines:

${\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.}$

Now we use law of cosines to find the second angle:

${\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.}$

Finally, β = 180° − α − γ.

Two sides and non-included angle given (SSA)Edit

Two sides and a non-included angle given

Two solutions for the triangle

This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines:

${\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .}$

We denote further D = c/b sin β (the equation's right side). There are four possible cases:

1. If D > 1, no such triangle exists because the side b does not reach line BC. For the same reason a solution does not exist if the angle β ≥ 90° and b ≤ c.
2. If D = 1, a unique solution exists: γ = 90°, i.e., the triangle is right-angled.
3. If D < 1 two alternatives are possible.
   1. If b ≥ c, then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique.
   2. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ′ = 180° - γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C′, side b′ and the angle γ′ as the second solution.

Once γ is obtained, the third angle α = 180° − β − γ.

The third side can then be found from the law of sines:

${\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}$

or

${\displaystyle a=c\cos \beta \pm {\sqrt {b^{2}-c^{2}\sin ^{2}\beta }}}$

A side and two adjacent angles given (ASA)Edit

One side and two adjacent angles given

The known characteristics are the side c and the angles α, β. The third angle γ = 180° − α − β.

Two unknown side can be calculated from the law of sines:

${\displaystyle a=c\ {\frac {\sin \alpha }{\sin \gamma }};\quad b=c\ {\frac {\sin \beta }{\sin \gamma }}.}$

or

${\displaystyle a=c{\frac {\sin \alpha }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}$

${\displaystyle b=c{\frac {\sin \beta }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}$