Use cosine ratio to calculate angles and sides (Cos = a / h)

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Introduction
Lessons
  1. What are sine, cosine and tangent?
Examples
Lessons
  1. Determine each cosine ratio using a calculator
    1. cos50\cos 50^\circ
    2. cos50\cos-50^\circ
  2. Determine the angle to the nearest degree using a calculator
    1. cosθ=0.24\cos \theta = 0.24
    2. cosθ=0.45 \cos \theta = -0.45
  3. Determine the angles and sides using cosine
    1. Find angle A and B:

      Use cosine ratio to calculate angle
    2. Find the value of "x" using cosine

      Use cosine ratio, adjacent side and cosine theta to find hypotenuse
    3. Find the value of " θ\theta " using cosine
      Use cosine ratio, adjacent and hypotenuse length to calculate cosine theta
  4. Draw and label a right triangle to illustrate Cosine ratio, then calculate the angle.
    1. cosθ=817\cos \theta ={8 \over 17}
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Topic Notes
Cosine ratios are exactly the same idea of sine ratios or tangent ratios. The only difference between it and the other two trigonometric ratios is that it is the ratio of the adjacent side to the hypotenuse of a right triangle.

Trigonometric ratios

cosine rule

The cosine rule tells us that when we have a right triangle, cosine=ahcosine = \frac{a}{h}. The "a" in this case stands for adjacent. The "h" stands for the hypotenuse, which can be found through the pythagorean theorem. In order to find cosine, all you'll need is the adjacent side and the hypotenuse.

sine rule

When you come across sine, you can find the answer for it in a right triangle by taking the opposite side over the hypotenuse in the form of sine=ohsine = \frac{o}{h}.

tangent rule

For the tangent rule, when you have a right triangle, you can use the opposite over the adjacent sides of the triangle to find your ratio. This means that tan=oatan = \frac{o}{a}.

SohCahToa

When you hear SohCahToa, it's not immediately obvious what it means. But it's actually an easier way for you to remember how to use sine, cosine, tangent that we just learned. These three are the main functions that you'll deal with in trigonometry problems.

Soh Cah Toa stands for:

Sine, opposite, hypotenuse, Cosine, adjacent, hypotenuse, Tangent, opposite, adjacent
Soh Cah Toa table

It can help you find the length of a side of a right triangle as long as you have an angle θ\theta and some info on the other sides of the triangle.

Example problems

In this chapter, we're actually going to focus on the cosine rule. This means we'll only be working with the "CahCah" portion of SohCahToaSohCahToa. Try out the following trig problems alongside us to learn how to solve questions using the cosine rule.

Question 1

Determine each cosine ratio using calculator

a) cos\cos 50°

Simply put the number into your calculator and you should get 0.640.64.

b) cos\cos -50°

Simply put the number into your calculator and you should get 0.640.64.

cos\cos 50° and cos\cos -50° both = 0.640.64. Why?

Chart of all, sine, tangent, cosine
ASTC chart

This above ASTC chart helps you figure out which trig ratio is positive in which quadrant. cos\cos 50° lies in quadrant I, where all the trig ratios are positive. cos\cos -50° lies in quadrant 4, where cosine is positive. This is why we get 0.640.64 for both cos\cos 50° and cos\cos -50°.

Question 2

Determine the angle to the nearest degree

Solution

a) cosθ=0.24\cos \theta = 0.24

Use the inverse cosine in the calculator which has a little 1-1 beside coscos:

arccos(0.24)=76\arccos (0.24) = 76°

b) cosθ=0.45\cos \theta = -0.45

Use the inverse cosine in the calculator to find:

arccos(0.45)=117\arccos (-0.45) = 117°

Question 3

Determine the angles and sides using cosine

Solution:

a) Find angle AA and BB:

Finding angle A and B with the information provided
Find angle A and B

cosθ=adjacenthypotenuse\cos \theta = \frac{adjacent}{hypotenuse}

Angle AA

cosA=817\cos A = \frac{8}{17}

To solve it in your calculator

arccos817=62\arccos \frac{8}{17} = 62°

Angle BB

cosB=1517\cos B = \frac{15}{17}

To solve it in your calculator

arccos1517=28\arccos \frac{15}{17} = 28°

b) Find the value of "xx" using cosine

Find the side of the triangle x using cosine
Find x using cosine/figcaption>

cosθ=adjacenthypotenuse\cos \theta = \frac{adjacent}{hypotenuse}

cos\cos 32° = 25(x)\frac{25}{(x)}

x=25cos32x = \frac{25}{\cos 32}

x=29x = 29

Still curious about the cosine rule? Try out this online law of cosines calculator!

Next up, you're going to learn more about the law of cosines, how to find exact trigonometric ratios, and touch on what the unit circle is in trigonometry.