Exam Response (trigonometric Functions)

Exam Response (trigonometric Functions)

What is it?

Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.

In simple words

• Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min.
• Write holes as x != formula.
• Amplitude is |b|, not c.
• A vertical shift (the +1) kills odd/even unless the shift is 0.

Important terms

EXAM RESPONSE (TRIGONOMETRIC FUNCTIONS): Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.

1. DOMAIN RANGE: sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.

2. GRAPH / PERIOD: sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.

What you should learn

• State the main ideas and notation used in exam response (trigonometric functions).
• Apply the rules of exam response (trigonometric functions) in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• csc 4x: k pi/4
• 7 cos 4x: [-7,7]
• tan: all R range

DOMAIN RANGE

sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.

GRAPH / PERIOD

sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.

Full school-level explanation

Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.

Domain/range, then graph words:

WHAT — EXAM RESPONSE (TRIGONOMETRIC FUNCTIONS)

Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.

RULE — 1. DOMAIN RANGE

sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.

METHOD — 2. GRAPH / PERIOD

sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.

WORKED EXAMPLE

Worked 4-mark: y = 1 + 2 sin(x/2). domain all reals range [1-2, 1+2] = [-1, 3] period 2 pi / (12) = 4 pi odd? 1+... is neither even nor odd

DOMAIN RANGE

sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.

Remember

• holes from sin=0 or cos=0

Examples

• csc 4x: k pi/4
• 7 cos 4x: [-7,7]
• tan: all R range

GRAPH / PERIOD

sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.

Remember

• T then amp

Examples

• sin(2x/3): T=3 pi
• 3 sin 2x: amp 3 T=pi
• tan unbounded

How to answer a exam response (trigonometric functions) question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• period sin(cx): 2 pi/|c|
• range a+b sin: [a-|b|, a+|b|]
• range [-1,3], T=4 pi
• shift => neither
• holes from sin=0 or cos=0
• T then amp
• period sin(cx): 2 pi/|c|
• range a+b sin: [a-|b|, a+|b|]

How the ideas connect

Worked 4-mark: y = 1 + 2 sin(x/2). domain all reals range [1-2, 1+2] = [-1, 3] period 2 pi / (12) = 4 pi odd? 1+... is neither even nor odd

Deeper understanding

Go beyond memorising Exam Response (trigonometric Functions). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked 4-mark: y = 1 + 2 sin(x/2). domain all reals range [1-2, 1+2] = [-1, 3] period 2 pi / (12) = 4 pi odd? 1+... is neither even nor odd

Common mistakes

Exam focus

Core Idea

1. DOMAIN RANGE

2. GRAPH / PERIOD

Worked Example

Exam focus

Quick revision

• period sin(cx): 2 pi/|c|
• range a+b sin: [a-|b|, a+|b|]
• range [-1,3], T=4 pi
• shift => neither
• EXAM RESPONSE (TRIGONOMETRIC FUNCTIONS): Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one perio
• 1. DOMAIN RANGE: sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin

A vertical shift (the +1) kills odd/even unless the shift is 0.

For long answers on Exam Response (trigonometric Functions): define clearly → explain the main parts → give one example → state one exam caution.