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# Sine, Cosine, Tangent

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Sine, Cosine, Tangent

What if you were given a right triangle and told that its sides measure 3, 4, and 5 inches? How could you find the sine, cosine, and tangent of one of the triangle's non-right angles? After completing this Concept, you'll be able to solve for these trigonometric ratios.

### Watch This

Watch the parts of the video dealing with the sine, cosine, and tangent.

### Guidance

Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used.

$& a \ \text{is} \ adjacent \ \text{to} \ \angle B \qquad \ a \ \text{is} \ opposite \ \angle A\\& b \ \text{is} \ adjacent \ \text{to} \ \angle A \qquad \ b \ \text{is} \ opposite \ \angle B\\& c \ \text{is the} \ hypotenuse$

The three basic trigonometric ratios are called sine, cosine and tangent. For right triangle $\triangle ABC$ , we have:

Sine Ratio: $\frac{opposite \ leg }{hypotenuse} \ \sin A = \frac{a}{c}$ or $\sin B = \frac{b}{c}$

Cosine Ratio: $\frac{adjacent \ leg}{hypotenuse} \ \cos A = \frac{b}{c}$ or $\cos B = \frac{a}{c}$

Tangent Ratio: $\frac{opposite \ leg}{adjacent \ leg} \ \tan A = \frac{a}{b}$ or $\tan B = \frac{b}{a}$

An easy way to remember ratios is to use SOH-CAH-TOA.

A few important points:

• Always reduce ratios (fractions) when you can.
• Use the Pythagorean Theorem to find the missing side (if there is one).
• If there is a radical in the denominator, rationalize the denominator.

#### Example A

Find the sine, cosine and tangent ratios of $\angle A$ .

First, we need to use the Pythagorean Theorem to find the length of the hypotenuse.

$5^2 + 12^2 &= c^2\\13 &= c\\\sin A &= \frac{leg \ opposite \ \angle A}{hypotenuse} = \frac{12}{13} && \cos A = \frac{leg \ adjacent \ to \ \angle A}{hypotenuse}= \frac{5}{13},\\\tan A &= \frac{leg \ opposite \ \angle A}{leg \ adjacent \ to \ \angle A}= \frac{12}{5}$

#### Example B

Find the sine, cosine, and tangent of $\angle B$ .

Find the length of the missing side.

$AC^2 + 5^2 &= 15^2\\AC^2 &= 200\\AC &= 10 \sqrt{2}\\\sin B &= \frac{10 \sqrt{2}}{15} = \frac{2 \sqrt{2}}{3} && \cos B = \frac{5}{15}=\frac{1}{3} && \tan B = \frac{10 \sqrt{2}}{5} = 2 \sqrt{2}$

#### Example C

Find the sine, cosine and tangent of $30^\circ$ .

This is a 30-60-90 triangle. The short leg is 6, $y = 6 \sqrt{3}$ and $x=12$ .

$\sin 30^\circ = \frac{6}{12} = \frac{1}{2} && \cos 30^\circ = \frac{6 \sqrt{3}}{12} = \frac{\sqrt{3}}{2} && \tan 30^\circ = \frac{6}{6 \sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

### Guided Practice

1. What is $\sin A$ ?

2. What is $\cos A$ ?

3. What is $\tan A$ ?

1. $\sin A=\frac{16}{20}=\frac{4}{5}$

2. $\cos A=\frac{12}{20}=\frac{3}{5}$

3. $\tan A=\frac{16}{12}=\frac{4}{3}$

### Explore More

Use the diagram to fill in the blanks below.

1. $\tan D = \frac{?}{?}$
2. $\sin F = \frac{?}{?}$
3. $\tan F = \frac{?}{?}$
4. $\cos F = \frac{?}{?}$
5. $\sin D = \frac{?}{?}$
6. $\cos D = \frac{?}{?}$

From questions 1-6, we can conclude the following. Fill in the blanks.

1. $\cos \underline{\;\;\;\;\;\;\;} = \sin F$ and $\sin \underline{\;\;\;\;\;\;\;} = \cos F$ .
2. $\tan D$ and $\tan F$ are _________ of each other.

Find the sine, cosine and tangent of $\angle A$ . Reduce all fractions and radicals.