Strategy’s

**Cosine law **

Cosine law is useful for finding the third side of a triangle when we know two sides and the angle between them (like the example above)

We know angle C = 37º, and sides a = 8 and b = 11

The law of cosine looks like: c^{2} = a^{2} + b^{2} − 2ab cos(C)

Step 1) Put in the known values – c^{2} = 8^{2} + 11^{2} − 2 × 8 × 11 × cos(37º)

Step 2) Do some calculations – c^{2} = 64 + 121 − 176 × 0.798…

Step 3) More calculations – c^{2} = 44.44…

Step 4) Take the square root – c = √44.44 = 6.67 to 2 decimal places

Answer)c = 6.67

You can also use Cosine Law to find the angles of a triangle when we know all three sides** (example below)**

The side of length “8” is opposite angle **C**, so it is side **c**. The other two sides are **a** and **b**.

Same as before, start with your basic equation – c^{2} = a^{2} + b^{2} − 2ab cos(C)

Step 1) put in the values – 8^{2} = 9^{2} + 5^{2 }− 2 × 9 × 5 × cos(C)

Step 2) calculate – 64 = 81 + 25 − 90 × cos(C)

Step 3) subtract the 25 from both sides – 39 = 81 − 90 × cos(C)

Step 4) Subtract 81 from both sides – −42 = −90 × cos(C)

Step 5) Swap sides – −90 × cos(C) = −42

Step 6) Divide both sides by −90 – cos(C) = 42/90

Step 7)Inverse cosine – C = cos^{-1}(42/90)

Step 8) Calculator – C = 62.2**°**

Trig ratios

Using a trig ratio is very helpful in solving for triangles. To solve a triangle means to find the length of all the sides and the measure of all the angles.

There are three steps:

Step 1) Choose which trig ratio to use.

– Choose either sin, cos, or tan by determining which side you know and which side you are looking for.

step 2) Substitute

– Substitute your information into the trig ratio.

Step 3) Solve

– Solve the resulting equation to find the length of the side.

1. Find b.

Step 1) Choose which trig ratio to use.

First, we know we must look at angle B because that is the angle we know the

measure of. So, looking at angle B, we want to identify which sides are involved. We know

one side is 8m, and that side is adjacent to angle B. The side we’re looking

for is opposite angle B. So we need to choose the trig ratio that has

opposite and adjacent. This of course is the tangent.

**Step 2) Substitute**

Next, we write our trig ratio: Tan B = opp/adj . Then, we substitute in the angle and the side we know: Tan 25 = b/8

Step 3) Solve

Now move the 8 to the other side by multiplying both sides by 8: 8 x tan 25 = b And use a calculator to find the answer. Well round to the nearest tenth: 3.7 m.

2. Find c.

Now that we know two sides, you could use the Pythagorean Theorem to find the third. But that’s less reliable because if you made a mistake on side b, then side c will also be wrong. So we are going to repeat the same process for side c.

Step 1) Choose the trig ratio to use.

We’re still using angle B. 8m is the adjacent and c is the hypotenuse**.** The trig ratio that uses the adjacent and hypotenuse is the cosine.

Step 2) Substitute

Write our trig ratio: Cos B = adj/hyp . Then, we substitute in the angle and the side we know: Cos 25 = 8/c

Step 3) Solve

Since our variable is on the bottom, we can start by cross multiplying: C x Cos 25 = 8 . Then we’ll divide both sides by cos 25°: C = 8/Cos 25 . And use a calculator to find the answer. Well round to the nearest tenth: 8.8 m.

**Sine law**

The Sine law says when we divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C

*a*sin A = *8*sin(62.2°) = *8*0.885… = 9.04…

*b*sin B = *5*sin(33.5°) = *5*0.552… = 9.06…

*c*sin C = *9*sin(84.3°) = *9*0.995… = 9.05…

Example

Law of Sines: a/sin A = b/sin B = c/sin C

Step 1) Enter values – a/sin A = 7/sin(35°) = c/sin(105°)

Ignore a/sin A (not applicable in this example) – 7/sin(35°) = c/sin(105°)

Step 2) swap sides – c/sin(105°) = 7/sin(35°)

Step 3) Multiply both sides by sin(105°) – c = ( 7 / sin(35°) ) × sin(105°)

Step 4) Calculate – c = ( 7 / 0.574… ) × 0.966…

Step 5) some more calculating – c= 11.8

we can also use the Law of Sines to find an unknown angle. In this case it is best to turn the fractions upside down (**sin A/a** instead of **a/sin A**, etc)

Step 1) enter the values – sin A / a = sin B / 4.7 = sin(63°) / 5.5

Step 2) Multiply both sides by 4.7 – sin B = (sin(63°)/5.5) × 4.7

Step 3) Calculate – sin B = 0.7614…

Step 4) Inverse Sine – B = sin^{-1}(0.7614…)

Answer) B = **49.6°**

There is one tricky thing we have to look out for: Two possible answers. Imagine we know angle **A**, and sides **a** and **b**.

We can swing side **a** to left or right and come up with two possible results (a small triangle and a much wider triangle) Both answers are right!