All solutions of trigonometric equations
Rearrange the ratio, use graph signs and include every solution in the stated interval.
Extended only.
Before you begin
- Use inverse trigonometric functions in degrees.
- Interpret the basic trig graphs.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Can sin x equal 1.2 for a real angle?
What you will learn
- Solve sine, cosine and tangent equations on 0° to 360°.
- Reject impossible ratios and check interval endpoints.
A calculator returns only a principal angle
For sin x = 0.6, inverse sine gives about 36.9°. Sine is also positive in the second quadrant, so the second solution is 180° − 36.9° = 143.1°, using the unrounded angle before final rounding. For cos x = −1/2, negative cosine occurs in quadrants two and three, giving 120° and 240°.
For tan x = −1, the solutions on a full turn are 135° and 315°, separated by the tangent period 180°. A negative calculator angle such as −45° must be moved into the stated interval. Mark the graph or the sign quadrants, then list all admissible solutions with degree units.
Rearrange first and check boundaries last
The equation 2 cos x + 1 = 0 becomes cos x = −1/2 before inverse calculation. If an equation instead requires sin x = 1.2, there is no real solution because sine lies between −1 and 1. An impossible input is a mathematical conclusion, not a calculator fault to ignore.
For sin x = 0 on the inclusive interval 0° ≤ x ≤ 360°, the solutions are 0°, 180° and 360°. Endpoints count when included by ≤. Tangent's asymptotes are never solutions because the function is undefined there. Substitute the final angles or use exact values to verify the original equation, not just the rearranged expression.
Pause and explain
Solve sin x = 0 on 0° ≤ x ≤ 360°.
Worked example
Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°.
Show the worked solution
- Rearrange to cos x = −1/2.
- Cosine is negative in quadrants two and three, with reference angle 60°.
- The solutions are 120° and 240°; substitution gives 2(−1/2) + 1 = 0.
Answer 120° and 240°.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve sin x = 1/2 on 0° ≤ x ≤ 360°.
Give me a hint
Sine is positive in two quadrants.
Compare my reasoning
- The reference angle is 30°.
- The second positive sine angle is 180 − 30 = 150°.
- Both 30° and 150° give sine one half in the interval.
30° and 150°.
Look for these in your work
- Included the second-quadrant solution.
- Checked the range.
Solve tan x = −1 on 0° ≤ x ≤ 360°.
Give me a hint
Use reference angle forty-five and the negative tangent quadrants.
Compare my reasoning
- The reference angle is 45°.
- Negative tangent occurs in quadrants two and four.
- Solutions are 180 − 45 = 135° and 360 − 45 = 315°.
135° and 315°.
Look for these in your work
- Used the correct sign quadrants.
- Listed both period-separated solutions.
Solve 3 sin x − 4 = 0 on a full turn. Explain why a decimal angle is not appropriate.
Give me a hint
Isolate sine before trying an inverse function.
Compare my reasoning
- The equation requires sin x = 4/3.
- Four thirds exceeds the maximum sine value one.
- There is no real solution in the interval or any other real-angle interval.
No real solution.
Look for these in your work
- Rearranged the original equation.
- Used the graph's range to justify the conclusion.
Common mistakes
- Keeping only the principal inverse angle.
- Missing inclusive interval endpoints.
- Ignoring an impossible sine or cosine ratio.
What can you explain now?
How far apart are tangent solutions with the same value?
Compare with the explanation
180° apart.
Tangent repeats after half a turn, so valid solutions recur with period 180 degrees.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, solve one equation for each trig function and explain the number of solutions from its graph.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
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