﻿ Sine, Cosine and Tangent ratios of a triangle. How to write the trig ratios of right triangles

# Sine, Cosine, Tangent Ratios

Practice writing the ratios

### Video Tutorial How to write sohcahtoa ratios, given side lengths

#### What do we mean by 'ratio' of sides?

Sine, cosine and tangent of an angle represent the ratios that are always true for given angles. Remember these ratios only apply to right triangles.

The 3 triangles pictured below illustrate this.

Although the side lengths of the right triangles are different, each one has a 37° angle, and as you can see, the sine of 37 is always the same!

In other words, sin(37) is always .6!

(Note: I rounded to the nearest tenth) That's, of course, why we can use a calculator to find these sine, cosine and tangent ratios.

### Practice Problems

Step 1

In the triangle above, what side is adjacent to $$\angle MLN$$?

Remember: When you are naming angles, the middle 'letter' or 'L' in this case is the angle in question.

The side adjacent to $$\angle M \red {L} N = ML$$

Step 2

What side is the hypotenuse?

LN

Step 3

Calculate $$cos(\angle MLN)$$

$$cos(\angle M\red{L}N) = \frac{adjacent}{hypotenuse}= \frac{8}{10} = .8$$

Step 4

Calculate $$cos(\angle MNL)$$

Remember: When you are naming angles, the middle 'letter' or 'L' in this case is the angle in question.

$$cos(\angle M\red{N}L) = \frac{6}{10} = .6$$

Remember that the sine ratio is the

$$\frac{\text{Opposite } }{ hypotenuse}. \\ sin(ACB) = \frac{opposite}{hypotenuse} = \frac{6}{10} = .6$$

Remember that the cosine ratio is the side adjacent to the angle (x) / hypotenuse.

$$cos(ACB) = \frac{adjacent}{hypotenuse}= \frac{4}{5} = .8$$

Remember that the tangent ratio is the side opposite /adjacent sides.

Therefore
$$tan(BAC) = \frac{opposite}{adjacent} \\ = \frac{24}{7} \\= 3.4285714285714284$$

$$\text{ for } \angle B\red{R}T \\ sin(\red {R})= \frac{opp}{hyp} = \frac{12}{13} = .923 \\ cos( \red {R})= \frac{adj}{hyp} = \frac{9}{13} = .69 \\ tan( \red {R})= \frac{opp}{adj} = \frac{12}{9} = 1.3$$

$$sin(Y\red{X}Z ) \frac{opposite}{hypotenuse} = \frac{24}{25} = .96$$

$$sin(Y\red{X}Z) = \frac{adjacent}{hypotenuse} = \frac{7}{25} = .28$$

$$sin(\angle G\red{H}I) = \frac{3}{5}= .6 \\ cos(\angle G\red{H}I) =\frac{ 4}{5} = .8 \\ tan(\angle G\red{H}I) = \frac{3}{4} = .75$$

$$tangent = \frac{opposite}{adjacent}$$

So which angle has a tangent that is equivalent to $$\frac{3}{4}$$?

$$\angle L \text { does } \\ tan(L) = \frac{9}{12}$$

$$cosine = \frac{adjacent}{hypotenuse}$$

So which angle has a cosine that is equivalent to $$\frac{3}{5}$$?

$$\angle K \\ cos(k) = \frac{9}{15}$$

This is a little trickier because you are given the ratio as a decimal; however, you only have two options. Either $$\angle A$$ or $$\angle C$$

$$tan( \angle A) = \frac{48}{14} \approx 3.42857 \\ \red{ tan( \angle C) = \frac{14}{48} \approx.29167 }$$

Again, there are two options (angles R or P), but since your ratio is greater than 1 you might quickly be able to notice that it must be R<

$$tan( \angle P) = \frac{5}{12} \approx 0. 41666 \\ \red{ tan( \angle R) = \frac{12}{5} = 2.4 }$$

Challenge Problem

Be careful! In the triangle on the left, which angle has a sine ratio of 2.6?

There is no angle that has a sine of 2.6.

Remember that the maximum value of the sine ratio is 1.

The same, is true, by the way of cosine ratio (max value is 1)

### Word Problems

Step 1

It's easiest to do a word problem like this one, by first drawing the triangle and labelling the sides. We know the opposite side of $$\angle K$$ and we know the hypotenuse

To get the tangent ratio we need to know the length of the adjacent side

How can we find the length of the adjacent side?

Step 2

Use the Pythagorean theorem!

$$a^2 + b^2 = c^2 \\ 3^2 + b^2 = 5^2 \\ b^2 = 5^2- 3^2 = 25-9 = 16 \\ b = 4$$

Now, use the tangent ratio!

Step 2

Use the Pythagorean theorem!

$$tan( k ) = \frac{opposite}{adjacent} \\ tan( k ) = \frac{3}{4}$$

Step 1

Draw this triangle and label the sides:

Remember that the cosine ratio = $$\frac{adjacent}{hypotenuse}$$.

How can we find the length of the opposite side?
(Remember that sine involves the opposite so we need to find that somehow)

Step 2

Use the Pythagorean theorem!

$$a^2 + b^2 = c^2 \\ 7^2 + b^2 = 25^2 \\ b^2 = 25^2- 7^2 = 625 - 49 = 576 \\ b = \sqrt{576} =24$$

Now, use the sine ratio!

Step 2

$$sin(b) = \frac{opposite}{hypotenuse}$$

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