Download trigonometric function

Warm Up
Given the measure of one of the acute
angles in a right triangle, find the
measure of the other acute angle.
1. 45° 45°
2. 60° 30°
3. 24° 66°
4. 38°
52°
Warm Up Continued
Find the unknown length for each right
triangle with legs a and b and hypotenuse
c.
5. b = 12, c =13
6. a = 3, b = 3
a=5
Objectives
Understand and use trigonometric
relationships of acute angles in
triangles.
Determine side lengths of right
triangles by using trigonometric
functions.
Vocabulary
trigonometric function
sine
cosine
tangent
cosecants
secant
cotangent
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.
The triangle shown at right is
similar to the one in the table
because their corresponding
angles are congruent. No
matter which triangle is used,
the value of sin θ is the same.
The values of the sine and
other trigonometric functions
depend only on angle θ and
not on the size of the triangle.
Example 1: Finding Trigonometric Ratios
Find the value of the sine,
cosine, and tangent functions
for θ.
sin θ =
cos θ =
tan θ =
Example 2
Find the value of the sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =
Example 3: Sports Application
In a waterskiing competition,
a jump ramp has the measurements shown. To
the nearest foot, what is
the height h above water
that a skier leaves the ramp?
Substitute 15.1° for θ, h for opp., and 19 for hyp.
Multiply both sides by 19.
Use a calculator to simplify.
5≈h
The height above the water is about 5 ft.
Caution!
Make sure that your graphing calculator is set to
interpret angle values as degrees. Press . Check
that Degree and not Radian is highlighted in the
third row.
Example 4 You Try!
A skateboard ramp will
have a height of 12 in.,
and the angle between
the ramp and the ground
will be 17°. To the nearest inch, what will be the length l of
the ramp?
Substitute 17° for θ, l for hyp., and 12 for opp.
Multiply both sides by l and divide by sin
17°.
Use a calculator to simplify.
l ≈ 41
The length of the ramp is about 41 in.
The reciprocals of the sine, cosine, and tangent ratios are also
trigonometric ratios. They are trigonometric functions, cosecant,
secant, and cotangent.
Example 5: Finding All Trigonometric
Functions
Find the values of the six trigonometric functions for
θ.
Step 1 Find the length of the hypotenuse.
a 2 + b 2 = c2
c2 = 242 + 702
c2 = 5476
c = 74
Pythagorean Theorem.
Substitute 24 for a and 70
for b.
Simplify.
Solve for c. Eliminate the
negative solution.
70
θ
24
Example 5 Continued
Step 2 Find the function values.
Helpful Hint
In each reciprocal pair of trigonometric functions, there is
exactly one “co”
Example 6 You Try!
Find the values of the six trigonometric functions for θ.
Step 1 Find the length of the hypotenuse.
a2 + b2 = c2
Pythagorean Theorem.
c2 = 182 + 802
Substitute 18 for a and 80
for b.
c2 = 6724
Simplify.
c = 82
Solve for c. Eliminate the
negative solution.
80
θ
18
Example 6 Continued
Step 2 Find the function values.
Lesson Quiz: Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric functions
for θ.