S
MDHS
MHF4U AM
Name: _______________________
Knowledge
Application
/10
/14
Thinking
19
Communication
/9
/7
Unit 4 Test - Trigonometric Equations
Knowledge [
/10]
14/9
1. Convert the 140 to radians. _______
𝑜
2. Convert
10π
9
[1]
2000
to degrees ________
[1]
3. How many possible solutions exist for 𝑠𝑖𝑛3𝑥 =
−1
2
6
in the interval 0 ≤ 𝑥 ≤ 2π? _________
[1]
D [1]
4. How many possible solutions exist for 𝑐𝑜𝑠𝑥 = 3 ? ___________
2 [1]
O T
5. Determine all solutions for 𝑠𝑖𝑛𝑥 = 0 in the interval 0 ≤ 𝑥 ≤ 2π ________________
,
,
I
[1]
( ) + 𝑠𝑖𝑛 ( ) ______________
2 2π
15
6. Evaluate the following using exact values: 𝑐𝑜𝑠
2 2π
15
7. Determine the exact value for each of the following: [2, 2]
3π
a) 𝑐𝑜𝑠 4
A
=-
T
cos(*(
+
)
I
g
π
6
b) 𝑐𝑠𝑐
=
Sin/6
=
=
1
=
2
t
#
Application [
/14]
8. Given 𝑡𝑎𝑛𝐴 =
7
4
2
where π < 𝐴 <
3π
2
:
a) Sketch the angle and determine the missing side length. [2]
A
S
p2
72442
=
49 + 16
=
r=
y
5
b) Determine the exact value of 𝑠𝑖𝑛2𝐴 [3]
-
=
=
=
2
Sin x
cost
~
-
2()(
S
~
S
9. Solve the following equations for when 0 ≤ 𝑥 ≤ 2π
a) 2𝑐𝑜𝑠𝑥 + 5 = 4
10. Choose one of the two identities to prove:
a) 𝑠𝑖𝑛𝑥 − 𝑡𝑎𝑛𝑦𝑐𝑜𝑠𝑥 =
𝑠𝑖𝑛(𝑥−𝑦)
𝑐𝑜𝑠𝑦
[4, 6]
b) 6𝑠𝑖𝑛2𝑥 = 3
[4]
4
b)
4
𝑐𝑜𝑠 𝑥−𝑠𝑖𝑛 𝑥
2
𝑠𝑖𝑛 𝑥
2
= 𝑐𝑜𝑡 𝑥 − 1
Thinking [
/9]
11. Solve 𝑐𝑜𝑠2𝑥 − 3𝑠𝑖𝑛𝑥 − 2 = 0 when 0 ≤ 𝑥 ≤ 2π
[5]
12. Choose one of the two identities to prove: [4]
2
a) 2𝑠𝑖𝑛 𝑥(1 + 𝑐𝑜𝑡𝑥) = 𝑠𝑖𝑛2𝑥 − 𝑐𝑜𝑠2𝑥 + 1
b)
𝑠𝑖𝑛2𝑥
2
2−2𝑐𝑜𝑠 𝑥
= 2𝑐𝑠𝑐2𝑥 − 𝑡𝑎𝑛𝑥
Communication [ /7]
13. A student thinks that 𝑠𝑖𝑛θ = 1 − 𝑐𝑜𝑠θ is an identity because 𝑠𝑖𝑛(0) = 1 − 𝑐𝑜𝑠(0) = 0.
Is this true or false? If false, provide a counterexample. [2]
Communication /5
- Form (equal signs, working down, proper “let” statements, therefore statements, etc)
- Grammar and Spelling (where applicable)
Trigonometric Identities
RECIPROCAL IDENTITIES:
PYTHAGOREAN IDENTITIES:
Compound Angle Identities
𝑡𝑎𝑛(𝐴 − 𝐵) =
QUOTIENT IDENTITIES:
Double Angle Identities
𝑡𝑎𝑛𝐴−𝑡𝑎𝑛𝐵
1+𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵