Download Ch. 5 Note Packet

Section 5.1 Analytic Trigonometry: Using Fundamental Identities
Section Objectives: Students will know how to use fundamental trigonometric identities to
Evaluate trigonometric functions and simplify trigonometric expressions.
Example 1: find the values of the other five trigonometric functions using the trigonometric identities.
(Note: Determine which quadrant u lies)
1a. If csc u = -5/3 and cos u > 0,
1b. If sec(-x) = -4, sinx = √ 28
4
1c. sec x is undefined, sin x < 0
5.1-1
Section 5.3 Solving Trigonometric Equations
Example 1. Solve the following on the interval [0, 2).
a) 2cos2 x + cos x – 1 = 0
b) 2cos2 x + 3sin x – 3 = 0
d) tan2x + 2 tanx = -1
e)
c) sec x + 1 = tan x
3tan2x – 1 = 0
SOLVE:
Example 2. Solve 1 – 2cos x = 0.
Example 3. Solve sin x + 1 = -sin x.
Example 4. Solve tan2 x – 3 = 0.
Example 5. Solve sec x csc x = csc x.
5.3-2
III. Functions Involving Multiple Angles (p. 361)
Example 6. Solve the multiple angle equations. Check with your calculator
a) 2sin 2t + 1 = 0
b) ) cot (x/2) + 1 = 0
c) sin 4 x 
2
.
2
IV. Example 7. Solve the multiple angle equation: tan23x = 3 for [0, 2π]
V. Using Inverse Functions (pp. 362 - 363)
Example 8. Solve over the interval [ 0, 2π ]
a) Solve sec2 x – 3sec x – 10 = 0.
b). Solve tan2x + 4tanx + 4 = 0
5.5-3
Section 5.4 Sum and Difference Formulas
Ex 1: Find the exact values: of the sine, cosine, and tangent of the angle
sin15º =
cos 15º =
tan 15º =
sin 7π =
12
cos 7π =
12
tan 7π =
12
5.5-4
Example 2.
Find the exact value of cos ( v – u ) given sin u = 5/13 and cos v = -3/5
with u & v in quad 2
Ex 3. Find the solutions of the equation in the interval [ 0, 2π ) .Use a graphing utility to verify
a) cos (x +  ) – cos (x - π ) = 1
4
4
b) 2sin ( x + π) + 3 tan ( π – x ) = 0
2
5.5-5
Section 5-5: Example 1: Use the figure to find the exact value of the trigonometric function
Sin x =
Cos 2x =
7
24
Cot 2x =
For Example 2 & 3: Find the exact solutions algebraically on the interval [ 0, 2π ) then check with a
graphing utility
Example 2: Solve 8 sinxcosx = 2
Example 3: Solve sin 4x – sin 2x = 0
Example 4: Given sin x = 12/ 13 and π/2 < x < π, find sin 2x, cos 2x, and tan 2x using your double angle
formulas
Sin 2x=
Cos 2x=
Tan 2x =
5.5-6
Half Angle formulas can also be helpful when keeping answers exact
Example 5: Use the figure to find the exact value of the trigonometric function
Sin x =
2
Sec x =
2
7
24
Cot x =
2
For Example 6a-c: Use the half-angle formula to find the exact values of the following
6a) cos 165º.
6b) tan 157º 30’
6c) sin 7π
12
Example 7: Using the half-angle formulas, find the exact values of sin u/2, cos u/2, tan u/2 when
cot u = 7 for π < u < 3π / 2
Sin u/2=
Cos u/2=
Tan u/2=
5.5-7
Example 8. Solve 2sin2 (x/2) = cos x on [0, 2). Use a graphing utility to check your answer
Example 9: Solve for cos 3x + cos x = 0 on [ 0, 2π ) . Use a graphing utility to verify your answer
5.5-8