12-2 Trigonometric Functions of Acute Angles

Trigonometric Functions

There are six trigonometric functions for any acute angle θ.

We have already discussed three:

r

y

hypotenuse

opposite sidesin

r

x

hypotenuse

adjacent sidecos

x

y

adjactent side

opposite sidetan

θ

side oppositey

side adjacent x

hypotenuse

r

x

y

Reciprocal Functions

The three remaining trigonometric functions are reciprocal functions of those previously defined.

The cotangent of θ, written cot θ, equals

The secant of θ, written sec θ, equals

The cosecant of θ, written csc θ, equals

y

x

tan

1

opposite side

adjacent side

x

r

cos

1

adjacent side

hypotenuse

y

r

sin

1

opposite side

hypotenuse

Finding Trigonometric Functions

Give the values of the six trigonometric functions of θ.

θ

3

7

x

y

Find the values of the six trigonometric functions of angle θ.

θ

9

15

x

y

Trigonometric Functions Using a Point

Find the values of the six trigonometric functions of an angle θ in standard position whose terminal side passes through (5, 12).

Trig IdentitiesAn equation involving trig function of an angle θ that is true for all values of θ is a trigonometric identity.

For example:

Pythagorean identity

tancos

sintan

cos

sin

x

y

rxry

θ

r

x

y

cotsin

coscot

sin

cos

y

x

ryrx

1cossin1cossin 222

2

2

22

2

2

2

22222

r

r

r

xy

r

x

r

y

r

x

r

y

Using Trig Identities

Find cos θ and tan θ if θ is an acute angle and .3

1sin

Find cos Φ if sin Φ = .13

5

Cofunctions

The sine and cosine are called cofunctions because sin A = cos B and sin B = cos A.

Similarly, sin A = cos (90 - A).

Other pairs of cofunctions: tangent and cotangent, secant and cosecant

sin θ = cos (90 - θ) cos θ = sin (90 - θ)

tan θ = cot (90 - θ) cot θ = tan (90 - θ)

sec θ = csc (90 - θ) csc θ = sec (90 - θ)

Using Cofunction Identities

Use the cofunction identities to find the measure of the acute angle θ.

• sin θ = cos 25

cot θ = tan 20.