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Sine, Cosine, and Tangent Functions

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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

CK-12 Foundation: The Trigonometric Ratios

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

James Sousa: Introduction to Trigonometric Functions Using Triangles

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}

CK-12 Foundation: The Trigonometric Ratios

Guided Practice

Answer the questions about the following image. Reduce all fractions.

1. What is \sin A ?

2. What is \cos A ?

3. What is \tan A ?

Answers:

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}

Practice

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.

Vocabulary

Acute Angle

Acute Angle

An acute angle is an angle with a measure of less than 90 degrees.
Adjacent Angles

Adjacent Angles

Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Hypotenuse

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
opposite

opposite

The opposite of a number x is -x. A number and its opposite always sum to zero.
Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.
Radical

Radical

The \sqrt{}, or square root, sign.
sine

sine

The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.
Trigonometric Ratios

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.

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