Further Trigonometric Identities and Equations
Further trigonometric identities and equations covers the identity sin^2(x) + cos^2(x) = 1 and tan(x) = sin(x)/cos(x), together with solving trig equations such as sin(x) = k, or quadratics in sin(x) or cos(x), over a range like 0 to 360 degrees. It extends GCSE trigonometry into exact values, identity proofs and multi-solution equations worth up to 7 marks in the AQA Level 2 paper.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the exact trig values for 0, 30, 45, 60 and 90 degrees so you can answer non-calculator questions.
- Use the identity sin^2(x) + cos^2(x) = 1 to find one ratio when given another, choosing the correct sign based on which quadrant or range the angle is in.
- Use tan(x) = sin(x)/cos(x) to connect the three ratios, or to rewrite an equation such as a sin(x) = b cos(x) as tan(x) = b/a.
- For linear equations like sin(x) = k, find the principal value using the inverse trig function, then use the symmetry of the graph (such as 180-x or 360-x) to find every solution in the given range.
- For quadratic trig equations, substitute s = sin(x) (or c = cos(x)) to form an ordinary quadratic, solve it, then convert each valid value back into angles.
- Check every solution lies within the given range before writing your final list of answers.
Worked example
Given that x is acute and cos(x) = 5/13, find the exact value of sin(x).
- Use sin^2(x) + cos^2(x) = 1, so sin^2(x) = 1 - (5/13)^2.
- (5/13)^2 = 25/169, so sin^2(x) = 1 - 25/169 = 144/169.
- Take the square root: sin(x) = sqrt(144/169) = 12/13.
- Since x is acute, sin(x) must be positive, so the positive root is correct.
Practice questions
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Q1Write down the exact value of cos(30 degrees).Show answer
Answer: sqrt(3)/2
Q2Write down the exact value of tan(60 degrees).Show answer
Answer: sqrt(3)
Q3Angle y is acute and cos(y) = 7/25. Find the exact value of sin(y).Show answer
Answer: 24/25 (using sin^2+cos^2=1: 25^2-7^2=576, sqrt(576)=24)
Q4Solve sin(x) = 0.7 for 0 <= x <= 360, giving your answers to 1 decimal place.Show answer
Answer: x = 44.4 or x = 135.6
Q5Solve 4cos^2(x) - 3 = 0 for 0 <= x <= 360, giving all solutions.Show answer
Answer: x = 30, 150, 210, 330 (cos^2(x) = 3/4, cos(x) = +-sqrt(3)/2)
Q6Solve 2sin^2(x) - sin(x) - 1 = 0 for 0 <= x <= 360, giving all solutions.Show answer
Answer: x = 90, 210, 330 (factorise (2sin(x)+1)(sin(x)-1)=0)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Solve tan(x) = -1.5 for 0 <= x <= 360, giving your answers to 1 decimal place.
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Solve 4cos^2(x) - 4cos(x) + 1 = 0 for 0 <= x <= 360, giving all solutions.
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Prove that (sin(x) + cos(x))^2 + (sin(x) - cos(x))^2 = 2 for all values of x.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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