Holt McDougal Algebra 2

Right-Angle TrigonometryRight-Angle Trigonometry

Holt Algebra 2Holt McDougal Algebra 2

• How do we understand and use trigonometric relationships of acute angles in triangles?

• How do we determine side lengths of right triangles by using trigonometric functions?

Holt McDougal Algebra 2

Right-Angle Trigonometry

A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ.

Holt McDougal Algebra 2

Right-Angle Trigonometry

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Trigonometric Ratios

1. Find the value of the sine, cosine, and tangent functions for θ.

sin θ =

cos θ =

tan θ =

opp.hyp.

adj.

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Trigonometric Ratios

2. Find the value of the sine, cosine, and tangent functions for θ.

sin θ =

cos θ =

tan θ =

opp.

hyp.adj.

Holt McDougal Algebra 2

Right-Angle Trigonometry

You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ration of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2.

Holt McDougal Algebra 2

Right-Angle Trigonometry

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Side Lengths of Special Right Triangles

3. Use a trigonometric function to find the value of x.

°

2x = 74

Which function relates the opposite and the hypotenuse?

Solve for x.

Substitute for sin 30°.

Substitute 30° for θ, x for opp, and 74 for hyp.

opp.

hyp.

x = 37

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Side Lengths of Special Right Triangles

4. Use a trigonometric function to find the value of x.

Which function relates the opposite and the hypotenuse?

opp.

hyp.

Substitute 45 for θ, x for opp, and 20 for hyp.

°°

Substitute for sin 45°.

Solve for x.2202 x

Holt McDougal Algebra 2

Right-Angle Trigonometry

Make sure that your calculator is set to interpret angle values as degrees. Press DRG. Check that Degree and not Radian is highlighted.

Caution!

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Side Lengths of Other Right Triangles

5. Use a trigonometric function to find the value of x.

opp.

adj.

Because x is on bottom, divide.o38tan

42x o38

42

x

Substitute 38 for θ, 42 for opp., and x for adj.

Which function relates the opposite and the adjacent?

x

4238tan o

76.53x

Holt McDougal Algebra 2

Right-Angle Trigonometry

Finding Side Lengths of Other Right Triangles

6. Use a trigonometric function to find the value of x.hyp.

adj.

Because x is on top, multiply.

o52cos155x

o52 155x

Substitute 52 for θ, x for adj., and 155 for hyp.

Which function relates the adjacent and the hypotenuse?

15552cos o x

43.95x

hyp

adjcos

Holt McDougal Algebra 2

Right-Angle Trigonometry

Lesson 10.1 Practice A