Chapter 7 Trigonometric Identities and Equations
7.1 Basic Trigonometric Identities
Identity – a statement that is ____________ for all values of 𝑥, 𝜃, 𝑒𝑡𝑐..
 Ex:
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Ex 1: Use Pythagorean Identities to find the trigonometric ratios algebraically.
5
3𝜋
a. sin 𝜃 = − 9 , 𝜋 ≤ 𝜃 ≤ 2 . Find cos 𝜃.
5
𝜋
b. tan 𝜃 = 3 , 0 ≤ 𝜃 ≤ 2 . Find cos 𝜃.
Ex 2: Find the reference angle of each radian value
28𝜋
a. 5
b.
c.
157𝜋
3
17𝜋
9
Ex 3: Express each value as a trigonometric function of an angle in Quadrant I.
28𝜋
a. cos 9
b. cot
25𝜋
c. sec
37𝜋
6
5
Ex 4: Simplify the following trigonometric expressions
cot 𝜃
a. tan 𝜃
b. cos 𝑥 + sin 𝑥 tan 𝑥
c. 𝑠𝑖𝑛2 𝜃 𝑐𝑜𝑠 2 𝜃 − 𝑐𝑜𝑠 2 𝜃
7.2 Verifying Trigonometric Identities
Verify Trig Identities
Hints:
Ex 1: Verify the following identities
cot 𝛼
a. cos 𝛼 = csc 𝛼
b. sin 𝛽 tan 𝛽 = sec 𝛽 − cos 𝛽
c. (𝑡𝑎𝑛2 𝑥 + 1)(𝑐𝑜𝑠 2 𝑥 − 1) = −𝑡𝑎𝑛2 𝑥
Ex 2: Find a numerical value of one trigonometric function of x.
1+tan 𝑥
a. 1+cot 𝑥 = 2
b. csc 𝑥 = sin 𝑥 tan 𝑥 + cos 𝑥
7.3 Sum and Difference Identities
Sum and Difference Identities
1.
2.
3.
Ex 1: Find the exact value of the following trigonometric functions
a. cos 195°
7𝜋
b. tan 12
c. csc
11𝜋
12
2
4
Ex 2: Find sin(𝛼 − 𝛽) when cos 𝛼 = 3 , cos 𝛽 = 5 . 𝛼 and 𝛽 are angles in Quadrant I.
4
3
Ex 3: Find sec(𝑥 + 𝑦) when sin 𝑥 = 7 , csc 𝑦 = 2. 𝑥 and 𝑦 are angles in Quadrant I.
Ex 4: Verify the following identities
a. cos(𝜋 − 𝜃) = − cos 𝜃
b. cos(𝐴 + 𝐵) =
1−tan 𝐴 tan 𝐵
sec 𝐴 sec 𝐵
7.4 Double-Angle and Half-Angle Identities
Double-Angle Identities
1.
2.
3.
4.
5.
6.
5
Ex 1: Given cos 𝛼 = 7 ,
3𝜋
2
≤ 𝛼 ≤ 2𝜋. Find sin 2𝛼 , cos 2𝛼 , tan 2𝛼.
Half-Angle Identities
1.
2.
3.
Ex 2: Find the exact value of sin 22.5°
5𝜋
Ex 3: Find the exact value of tan 12
Ex 4: Verify the following identities
cos 2𝑥−1
a. cos 𝑥 − 1 = 2(cos 𝑥+1)
b. 1 − cos 2𝛼 sec 2 𝛼 = tan2 𝛼
2.1 Solving Systems of Equations in Two Variables
Definitions
 System of equations – a set of _________ or more equations
o Ex:

Solution – the ________________________ of the graphs represents the point at which the equations
have the same x – value and the same y – value
o 3 types:
Ex 1: Solve the system of equations by graphing
3𝑥 − 2𝑦 = −6
𝑥 + 𝑦 = −2
Ex 2: Use elimination to solve the following system of equations
1.5𝑥 + 2𝑦 = 20
2.5𝑥 − 5𝑦 = −25
Ex 3: Use substitution to solve the system of equations
a. 2𝑥 + 3𝑦 = 8
𝑥−𝑦=2
b. 𝑥 − 𝑦 = 2
2𝑥 = 2𝑦 + 10
7.5 Solve Trigonometric Equations
Goal:
READ DIRECTIONS CAREFULLY!!!!!
Ex 1: Solve for principal values of x
a. −2 sin 𝑥 = √3
b. √3 tan 𝑥 = 1
Ex 2: Solve for x when 0 ≤ 𝑥 ≤ 2𝜋.
a. cos 𝑥 tan 𝑥 = −
√3
2
b. 2 cos2 𝑥 + 3 cos 𝑥 = 2
Ex 3: Solve for all values of x
a. sin2 2𝑥 + cos 2 𝑥 = 0
b. sin2 𝑥 − sin 𝑥 + 1 = cos2 𝑥
7.6 Normal Form of a Line
Normal line – a line that is ____________________________ to another line, curve or surface
 Given a line in the xy-plane, there is a normal line that intersects the given line and passes through the
__________________.
 Ex:
Normal form of a line
Ex 1: Write the equation of a line in standard form when the length of the normal is 7 and 𝜑 =
Going from Standard Form to Normal Form
Ex 2: Rewrite 2𝑥 + 4𝑦 + 8 = 0 into normal form
5𝜋
6
3
1
Ex 3: Rewrite 𝑦 = 4 𝑥 − 5 into normal form
3
1
Ex 4: Find the length of normal and 𝜑 for the line 𝑦 = 4 𝑥 − 5
Ex 5: Find 𝜑 to the nearest hundredth given normal form 0 = −
√65
𝑥
65
+
8√65
65
𝑦−
72√65
65
7.7 Distance Between a Point and a Line
2
Ex 1: Find the exact distance between (7,6) and the line 𝑦 = − 3 𝑥 − 𝑦 + 7
Ex 2: Find the distance between 𝑦 = 3𝑥 − 11 and 𝑦 = 3𝑥 + 2