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Five-Minute Check (over Chapter 4)
Then/Now
New Vocabulary
Key Concept: Reciprocal and Quotient Identities
Example 1: Use Reciprocal and Quotient Identities
Key Concept: Pythagorean Identities
Example 2: Use Pythagorean Identities
Key Concept: Cofunction Identities
Key Concept: Odd-Even Identities
Example 3: Use Cofunction and Odd-Even Identities
Example 4: Simplify by Rewriting Using Only Sine and Cosine
Example 5: Simplify by Factoring
Example 6: Simplify by Combining Fractions
Example 7: Rewrite to Eliminate Fractions
Over Chapter 4
Find the exact values of the six
trigonometric functions of θ.
A.
B.
C.
D.
Over Chapter 4
If
, find the exact values of the five
remaining trigonometric function values of θ.
A.
B.
C.
D.
Over Chapter 4
Write −150° in radians as a multiple of π.
A.
B.
C.
D.
Over Chapter 4
Solve ∆ABC if A = 33°, b = 9, and c = 13.
Round side lengths to the nearest tenth and
angle measures to the nearest degree.
A. B = 42°, C = 105°, a = 7.3
B. B = 52°, C = 95°, a = 9
C. B = 95°, C = 52°, a = 7.3
D. B = 105°, C = 42°, a = 7.3
Over Chapter 4
Find the exact value of
A.
B.
C.
D.
You found trigonometric values using the unit circle.
(Lesson 4-3)
• Identify and use basic trigonometric identities to find
trigonometric values.
• Use basic trigonometric identities to simplify and
rewrite trigonometric expressions.
• identity
• trigonometric identity
• cofunction
• odd-even identities
Use Reciprocal and Quotient Identities
A. If
, find sec θ.
Reciprocal Identity
Divide.
Answer:
Use Reciprocal and Quotient Identities
B. If
and
, find sin x.
Reciprocal Identity
Quotient Identity
Substitute
for cos x.
Use Reciprocal and Quotient Identities
Divide.
Multiply each side by
Simplify.
Answer:
.
If
A.
B.
C.
D.
, find sin  .
Use Pythagorean Identities
If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
Use the Pythagorean Identity that involves cot θ.
cot 2 θ + 1 = csc 2 θ
Pythagorean Identity
(2) 2 + 1 = csc 2 θ
cot θ = 2
5 = csc 2 θ
Simplify.
= csc θ
Take the square root of
each side.
Reciprocal Identity
Solve for sin θ.
Use Pythagorean Identities
Since
is positive and cos θ < 0, sin θ
must be negative. So
. You can
then use this quotient identity again to find cos θ.
Quotient Identity
cot θ = 2 and
Multiply each side by
.
Use Pythagorean Identities
So,
Answer:
Check
sin 2 θ + cos 2 θ = 1
Pythagorean Identity

Simplify.
Find the value of csc  and cot  if
cos < 0.
A.
B.
C.
D.
and
Use Cofunction and Odd-Even Identities
If cos x = –0.75, find
Factor.
Odd-Even Identity
Cofunction Identity
cos x = –0.75
Simplify.
Use Cofunction and Odd-Even Identities
So,
= 0.75.
Answer: 0.75
If cos x = 0.73, find
A. –0.73
B. –0.68
C. 0.68
D. 0.73
.
Simplify by Rewriting Using Only Sine and
Cosine
Simplify
.
Solve Algebraically
Pythagorean Identity
Multiply.
Simplify.
So,
= cos x.
Simplify by Rewriting Using Only Sine and
Cosine
Support Graphically
The graphs of
appear to be identical.
Answer: cos x
and y = cos x
Simplify csc x – cos x cot x.
A. cot x
B. tan x
C. cos x
D. sin x
Simplify by Factoring
Simplify cos x tan x – sin x cos 2 x.
Solve Algebraically
cos x tan x – sin x cos 2 x
Original expression
Quotient Identity
Multiply.
Factor.
Pythagorean Identity
= sin3x
Simplify.
So, cos x tan x – sin x cos 2 x = sin3 x.
Simplify by Factoring
Support Graphically
The graphs below appear to be identical.
Answer: sin3 x
Simplify cos2 x sin x – cos(90° – x).
A. –sin3 x
B. sin3 x
C. cos2 x – 1
D. sin x cos x
Simplify by Combining Fractions
Simplify
.
Common
denominator
Multiply.
Add the
numerators.
Simplify.
Pythagorean
Identity
Simplify by Combining Fractions
Reciprocal
Identity
Reciprocal
and Quotient
Identities
Divide out
common factor.
Reciprocal
Identity
–2csc2 x.
Answer: – 2 sec2 x – 2 sec2 x
Simplify
A. cos x
B. 2 + 2 cos x
C. 2 sin x
D. 2 csc x
.
Rewrite to Eliminate Fractions
Rewrite
as an expression that does not
involve a fraction.
Pythagorean Identity
Reciprocal Identity
Reciprocal Identity
Quotient Identity
Rewrite to Eliminate Fractions
So,
= tan2 x.
Answer: tan 2 x
Rewrite
involve a fraction.
A. –2 tan2 x
B. 1+ sin x
C. 1 – cos x
D. 2 sin x
as an expression that does not