Download Study Guide and Intervention

NAME
DATE
5-3
PERIOD
Study Guide and Intervention
Solving Trigonometric Equations
Use Algebraic Techniques to Solve To solve a trigonometric
equation, you may need to apply algebraic methods. These methods include
isolating the trigonometric expression, taking the square root of each side,
factoring and applying the Zero-Product Property, applying the quadratic
formula, or rewriting using a single trigonometric function. In this lesson,
we will consider conditional trigonometric equations, or equations that may
be true for certain values of the variable but false for others.
Example 1
Find all solutions of tan x cos x - cos x = 0 on the
interval [0, 2π).
tan x cos x - cos x = 0
Original equation
cos x (tan x - 1) = 0
Factor.
cos x = 0 or
tan x - 1 = 0
Set each factor equal to 0.
3π
π
x=−
or −
2
tan x = 1
2
5π
π
x=−
or −
4
4
3π
π
or −
, tan x is undefined, so the solutions of the original
When x = −
2
2
5π
π
or −
. When you solve for all values of x, the solution should
equation are −
4
4
be represented as x + 2nπ for sin x and cos x and x + nπ for tan x, where n
5π
π
+ nπ or −
+ nπ.
is any integer. The solutions are −
4
Example 2
4
Find all solutions of sin x + √
3 = -sin x.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3 = -sin x
sin x + √
√
2 sin x + 3 = 0
2 sin x = - √
3
Original equation
Add sin x to each side.
Subtract √
3 from each side.
√3
Divide each side by 2.
2
5π
4π
x=−
or −
Solve for x.
3
3
5π
4π
The solutions are −
+ 2nπ or −
+ 2nπ.
3
3
sin x = - −
Exercises
Solve each equation for all values of x.
1. cos x = -1
2. sin3 x - 4 sin x = 0
3. sin x cos x -3 cos x = 0
4. 2 sin3 x = sin x
Find all solutions of each equation on the interval [0, 2π).
5. 2 cos x = 1
6. 5 + 2 sin x - 7 = 0
7. 4 sin2 x tan x = tan x
8. 2 cos x - √
3 =0
Chapter 5
16
Glencoe Precalculus
NAME
5-3
DATE
Study Guide and Intervention
PERIOD
(continued)
Solving Trigonometric Equations
Use Trigonometric Identities to Solve You can use trigonometric
identities along with algebraic methods to solve trigonometric equations. Be
careful to check all solutions in the original equation to make sure they are
valid solutions.
Example 1
Find all solutions of 2 tan2 x - sec2 x + 3 = 1 - 2 tan x
on the interval [0, 2π).
2 tan2 x - sec2 x + 3 = 1 - 2 tan x
Original equation
2
2
2 tan x - (tan x + 1) + 3 = 1 - 2 tan x
sec2 x = tan2 x + 1
tan2 x + 2 = 1 - 2 tan x
Simplify.
2
tan x + 2 tan x + 1 = 0
Simplify.
2
(tan x + 1) = 0
Factor.
tan x = -1
Take the square root of each side.
3π
7π
x = − or −
Solve for x on [0, 2π).
4
Example 2
Find all solutions of 1 + cos x = sin x on the interval
[0, 2π).
1 + cos x = sin x
Original equation
2
2
(1 + cos x) = (sin x)
Square each side.
2
2
Multiply.
1 + 2 cos x + cos x = sin x
2
2
1 + 2 cos x + cos x = 1 - cos x
Pythagorean Identity
2
2 cos x + 2 cos x = 0
Simplify.
2 cos x (cos x + 1) = 0
Factor.
cos x = 0 or cos x = -1
Zero Product Property
3π
π
x = −, π, −
Solve for x on [0, 2π).
2
Lesson 5-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
2
Exercises
Solve each equation for all values of x.
1. tan2 x = 1
2. 2 sin2 x - cos x = 1
3. sin x cos x -3 cos x = 0
4. cos2 x + sin x + 1 = 0
Find all solutions of each equation on the interval [0, 2π).
5. cos x = sin x
6. √
3 cos x tan x - cos x = 0
7. tan2 x + sec x - 1 = 0
8. 1 + cos x = √
3 sin x
Chapter 5
17
Glencoe Precalculus
NAME
5-3
DATE
PERIOD
Practice
Solving Trigonometric Equations
Solve each equation for all values of x.
1. cos x = 3 cos x - 2
3.
√
cos x
= 2 cos x - 1
2. 2 sin2 x - 1 = 0
4. 2 sin2 x - 5 sin x + 2 = 0
Find all solutions of each equation on the interval [0, 2π).
5. sec2 x + tan x = 1
6. 3 tan x - √
3=0
7. 4 sin2 x - 4 sin x + 1 = 0
8. 4 cos2 x - 1 = 0
3
x
−
9. cos
= cot x
sin x
10. tan x sin2 x = 3 tan x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11. CIRCLES To find the diameter d of any circle, first inscribe a triangle in
the circle. The diameter is then equal to the ratio of any side of the
triangle and the sine of its opposite angle.
a. Suppose the measure of one side of a triangle inscribed in a circle is
20 centimeters. If the measure of the angle in the triangle opposite
this side is 30°, what is the length of the diameter of the circle?
b. Suppose a circle with a diameter of 12.4 inches circumscribes a
triangle with one side of the triangle measuring 4.6 inches. What is
the measure of the angle in the triangle opposite this side?
12. AVIATION An airplane takes off from the ground and reaches a height
of 500 feet after flying 2 miles. Given the formula H = d tan θ, where H
is the height of the plane and d is the distance (along the ground) the
plane has flown, find the angle of ascent θ at which the plane took off.
Chapter 5
18
Glencoe Precalculus