Trig Identities & Reduction Formulae: Full Guide — Worksheet & Memo

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Grade 11 · Trigonometry

Trig Identities & Reduction Formulae: Full Guide

MEMORANDUM

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Class:

Date:

Mark

/ 23

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Simplify sinθ · tanθ · cosθ, and state the restriction on θ. (simplify)
(3)
sinθ·sinθ/cosθ·cosθ = sin²θ ✓✓
The original expression contains tanθ, so cosθ ≠ 0, i.e. θ ≠ 90° + k·180° ✓(3)
2
Simplify (1 − sin²θ). (identity)
(1)
cos²θ✓ (1)
3
Simplify cos²θ/(1 − sin²θ). (identity)
(2)
cos²θ/cos²θ = 1✓✓ (2)
4
Prove tanθ · cosθ = sinθ, and say where the identity does not hold. (prove)
(3)
tanθ = sinθ/cosθ ✓, so sinθ/cosθ·cosθ = sinθ ✓
tanθ is undefined where cosθ = 0, so the identity does not hold at θ = 90° + k·180° ✓(3)
5
In which quadrant are only sine positive? (CAST)
(1)
2nd quadrant✓ (1)
6
sinθ = 0,5, θ in Q2. Is cosθ + or −? (CAST)
(1)
Q2 → cosine negative✓ (1)
7
Simplify cos(180° − θ). (reduction)
(1)
−cosθ✓ (1)
8
Simplify sin(360° − θ). (reduction)
(1)
−sinθ✓ (1)
9
Exact value of sin 150°. (exact)
(2)
150° = 180° − 30°, Q2 sine + → sin 30° = 1/2✓✓ (2)
10
Exact value of cos 210°. (exact)
(2)
210° = 180° + 30°, Q3 cosine − → −cos 30° = (−√3)/2✓✓ (2)
11
For which values of θ is tanθ undefined? Give the general answer.
(2)
θ = 90° + k · 180°, k ∈ ℤ ✓✓ (cosθ = 0 there, and tanθ = sinθ/cosθ)(2)
12
For θ ∈ [0° ; 360°], state the values for which (sinθ × tanθ)/cosθ is undefined.
(2)
cosθ = 0, so θ = 90° and θ = 270° ✓✓(2)
13
Explain why the restriction must be read off the ORIGINAL expression and not the simplified one.
(2)
Simplifying can cancel the factor that caused the division by zero ✓; the original expression is still undefined there, so the restriction stands ✓.(2)

TOTAL: 23 marks