LEARNING OBJECTIVES
What you will be able to do
- use all six trigonometric functions for angles of any magnitude
- interpret amplitude, period and graph transformations
- solve trigonometric equations over degree or radian domains
- prove identities using reciprocal and Pythagorean relationships
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Connect six functions, transformed graphs and identities so equations can be solved over any stated domain.
Additional Mathematics extends trigonometry beyond right triangles. Signs, periodicity and exact relationships allow the same function value to occur at many angles.
A complete solution respects the stated domain. When the argument is bx rather than x, transform the domain before listing angles and divide back only at the end.
SECTION 01
Six functions and exact relationships
Reference angles give magnitudes, while the quadrant gives the sign. Exact special-angle values are often expected on the non-calculator paper.
| Quadrant | Positive functions |
|---|---|
| I | all six |
| II | sin and cosec |
| III | tan and cot |
| IV | cos and sec |
SECTION 02
Graphs, amplitude and period
For y = a sin bx + c or y = a cos bx + c, amplitude is |a| and the centre line is y = c. Sine and cosine periods are 2π/|b| radians; tangent has period π/|b| and vertical asymptotes.
DETAILED EXPLANATION
- A vertical translation moves a tangent graph up or down but leaves its vertical asymptotes at the same x-values.
- Label tangent asymptote x-coordinates.
- The graph domain may be given in radians or degrees.
SECTION 03
Equations over a domain
Rewrite reciprocal functions when helpful, reduce the equation to one trig function, find a reference angle, then list every solution in the transformed domain.
Solve with a doubled angle
- Solve cos(2x) = −1/2 for 0 ≤ x ≤ 180°.
- The transformed domain is 0 ≤ 2x ≤ 360°.
- Cosine is −1/2 at 2x = 120° and 240°.
- Divide both values by 2.
Answer: x = 60° or 120°.
SECTION 04
Identities and proofs
An identity is true for all values where both sides are defined. Work from the more complicated side, convert reciprocals to sine and cosine when useful, factor or combine fractions, and finish at the other side.
DETAILED EXPLANATION
- Manipulate one side rather than assuming the desired result.
- State restrictions if a denominator can be zero.
- Factor before cancelling.
- A proof needs an unbroken chain of equivalent statements.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Secant, cosecant and cotangent are reciprocal functions.
- Quadrants determine signs and periods generate repeated solutions.
- Amplitude and period come from coefficients in the function.
- Transform the domain when the angle is multiplied.
- Identity proofs use exact algebra and standard relationships.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- use all six trigonometric functions for angles of any magnitude
- interpret amplitude, period and graph transformations
- solve trigonometric equations over degree or radian domains
- prove identities using reciprocal and Pythagorean relationships