Trigonometric Ratios and Their Inverses

is the study of relationships between sides of right triangles and their angle measurements.

Def: Trigonometry

The Sides in Relation to an Angle

θ

OppositeHypotenuse

Adjacent

θ =is Greek for “Theta”

Now you try…Which side is opposite, adjacent, and the hypotenuse to give the angle?

1. Angle X 2. Angle Y f

e

d

Y

X

Answer: Angle X-d=opposite, e=adjacent, f=hypotenuse

Angle Y-e=opposite, d=adjacent, f=hypotenuse

3 Basic Trigonometric Ratios1. The Sine Ratio

– also known as sin

θ

OppHyp

Adj

9

7

15

6.05

3

15

9sin

?sinisWhat

2. The Cosine Ratio – also known as cos

θ

OppHyp

Adj

9

7

15

?cosisWhat

467.015

7cos

3. The Tangent Ratio – also known as tan

θ

OppHyp

Adj

9

7

15

?tanisWhat

286.17

9tan

A Mnemonic Device

Using Trig to Find Missing SidesExample:Steps1. Decide which trig ratio you need.

A

7 mC B

56°

2. Write an equation

3. Use calculator to find trig value. 4. Cross multiply to solve the equation.

Now you try…Find the missing side: 1. 2.

3. a

27°

4 cmz

31°

10 cm

b

15 m

48°

Inverse Trigonometry • Sine, Cosine, and Tangent are used to find

missing sides of a right triangle.

• The inverses of these ratios are used to find __________ of a right triangle.

Notation: 1. Inverse Sine: or 2. Inverse Cosine: or 3. Inverse Tangent: or

Example:Steps1. Decide which trig ratio you need.

2. Write an equation

3. Take the inverse of each side of the equation.

4. Use calculator to find inverse trig value. This value is the angle measurement.

Now you try…

1. Find angle R. 2. Find angle A.

A

C

B

7 m

14 m

3. Find θ Homework: Pg. 624 #1-19