Side Relationships in Special Right Triangles &

Exact Values of The Trigonometric Functions

Side Relationships in Special Right Triangles

The 45° – 45 ° – 90 ° Theorem

In a 45° - 45° - 90° triangle the hypotenuse

is times as long as either leg.

5 2

x

45°

Example: Find the value of x in the triangle.

hyp = leg

hyp = 5

Therefore, x =

2

2

2

Side Relationships in Special Right Triangles

The 30° – 60 ° – 90 ° Theorem

In a 30° - 60° - 90° triangle the hypotenuse is 2

times as long as the shorter leg. The longer leg

is times as long as the shorter leg.

10 3x

3

3

3

x30°

y60°

20

Example: Find the value of x and y in the triangle.

hyp = 2 shorter leg longer leg = shorter leg

20 = 2 y x = 10

10 = y

Exact Values of The Trigonometric Functions

There are certain angles for trigonometric functions which have exact values

0°, 90°, 180°, 270°, 30°, 45°, and 60°

0° 30° 45° 60° 90°

Sin 0 1

Cos 1 0

Tan 0 1 undefined

3

2

3

2 3

1

2

1

2

2

2

2

2

3

2

Exact Values of The Trigonometric Functions

Example: Find the exact values of the six

trigonometric functions of 0°.

Solution: Begin by drawing an angle of 0°, and

choose any point that would lie on the terminal

side. (3, 0) is one such point.

x = 3

y = 0

r = 3(3,0)

x

y

0 3 0sin 0 cos0 tan 0

3 3 33 3 3

csc0 sec0 cot 00 3 0undefined undefined

Homework

Do #1 – 7 odd numbers only on page 235 from Section 7.4 and #1 and 2 on page 238 from Section 7.5 for Tuesday June 9th