Higher Maths · Expressions and Functions
Trigonometric Functions | Higher Maths
Trigonometric functions in Higher Maths require fluency with radians, exact values for special angles and the basic identities. Trig appears in nearly every Higher exam, often as a multi-step Paper 2 question.
Core identities
sin2x + cos2x = 1
tan x = sin x / cos x
Radians: 180° = π radians
Conversions: x° × (π/180) to radians; x rad × (180/π) to degrees.
Exact values to memorise
sin 30° = 1/2 ; cos 30° = √3/2 ; tan 30° = 1/√3
sin 45° = cos 45° = 1/√2 ; tan 45° = 1
sin 60° = √3/2 ; cos 60° = 1/2 ; tan 60° = √3
Worked example
Worked example — Solving a trig equation
Problem: Solve 2 sin x − 1 = 0 for 0 ≤ x ≤ 2π.
- Rearrange.sin x = 1/2
- Use exact value: sin x = 1/2 when x = π/6.But sine is positive in the 1st and 2nd quadrants.
- Find the second solution using x = π − π/6.x = 5π/6
- State both solutions.x = π/6 or 5π/6
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Solve cos x = −√3/2 for 0 ≤ x ≤ 2π.
Show answer
Cosine negative in 2nd and 3rd quadrants. x = 5π/6 or 7π/6 - Convert 135° into radians.
Show answer
135 × π/180 = 3π/4 - Simplify (1 − cos2x) / sin x.
Show answer
= sin2x / sin x = sin x - Solve 3 tan x − √3 = 0 for 0 ≤ x < 2π.
Show answer
tan x = √3/3 = 1/√3 ⇒ x = π/6 or π + π/6. x = π/6 or 7π/6 - Show that sin2x − cos2x = 1 − 2cos2x.
Show answer
LHS = (1 − cos2x) − cos2x = 1 − 2cos2x = RHS.
Common mistakes
Common mistakes & how to avoid them
- Working in degrees when the question states radians (or vice versa) — always check the limits.
- Forgetting that sin x = k has two solutions in [0, 2π] (in different quadrants).
- Not memorising the exact values — the SQA formula sheet does not contain them.
Frequently asked questions
Are trig identities on the formula sheet?
sin2x + cos2x = 1 is not on the SQA Higher formula sheet, but tan x = sin x / cos x is implied. Memorise both.
Why does Higher Maths use radians instead of degrees?
Because differentiation and integration of trig functions only work cleanly when the angle is in radians.
What does CAST help me remember?
CAST tells you which trig ratios are positive in each quadrant: 4th = Cos, 1st = All, 2nd = Sin, 3rd = Tan. Use it to find all solutions in [0, 2π].
Related Higher Maths topics
These topics often appear together in SQA exam questions.
Need one-to-one help with Trigonometric Functions?
Our Glasgow Higher Maths specialists run 1-hour one-to-one sessions and 80-minute group classes — online or at our Pollokshields Tuition Centre.
Ready to enrol?
Get matched with a Glasgow specialist
Tell us the subject and level — we’ll come back the same day with availability and a clear quote.