Basic Trig Equations

Key idea

Solve sinθ = k using reference angles and periodicity.

In depth

Solve sin x = c by finding a reference angle and using the quadrants where the sign is right, then adding the period to get all solutions. Always state the interval required.

Worked example

sin x = 1/2 on [0, 2π]: x = π/6 and 5π/6.

Real-world uses

• Finding times when a tide is at a certain height or when a spring is at a set position

Common mistake

Giving only one solution when there are two in each period.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
• Pythagorean Identity – The key identity from the unit circle.