Pure: Trigonometry
Trigonometry at A Level extends GCSE work to radians, exact trig values, the sine and cosine rules, small-angle approximations, and identities including the double angle formulae, then uses them to solve trig equations over a given range. It underpins mechanics and calculus, since differentiating and integrating trig functions relies on these identities.
Method
- Convert between degrees and radians when needed, using pi radians = 180 degrees; arc length = r*theta and sector area = 0.5*r^2*theta only work with theta in radians.
- For triangles without a right angle, use the sine rule (a/sin(A) = b/sin(B) = c/sin(C)) for missing sides or angles, or the cosine rule (a^2 = b^2 + c^2 - 2bc*cos(A)) when you know two sides and the included angle, or three sides.
- Learn the exact trig values for 30, 45, 60, 90 and 180 degrees, since exam questions often expect exact surd answers rather than decimals.
- Use trig identities (such as sin^2(x) + cos^2(x) = 1, tan(x) = sin(x)/cos(x), and the double angle formulae) to simplify an equation into a single trig function before solving.
- When solving trig equations over a given range, find the principal solution with an inverse trig function, then use the symmetry of the sine, cosine or tangent graph to find every other solution in the range.
- Check every solution lies within the range given in the question, and reject any that fall outside it or that came from an invalid step.
Worked example
Solve 2sin(2x) = 1.5 for 0 degrees <= x <= 180 degrees, giving your answers to 1 decimal place.
- Rearrange to sin(2x) = 0.75.
- Let u = 2x, so the range for u is 0 to 360 degrees, and solve sin(u) = 0.75.
- Find the principal value: u = arcsin(0.75) = 48.6 degrees (1dp).
- Sine is also positive in the second quadrant, so the second value is u = 180 - 48.6 = 131.4 degrees.
- No other values of u fall in the range 0 to 360, so u = 48.6 or u = 131.4 degrees.
- Final answer: divide both values by 2 to return to x: x = 24.3 degrees or x = 65.7 degrees.
Practice questions
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Q1Convert 3pi/4 radians to degrees.Show answer
Answer: 135 degrees.
Q2A sector has radius 6 cm and angle 2 radians. Find the arc length.Show answer
Answer: 12 cm (s = r x theta = 6 x 2).
Q3Find the exact value of cos(150 degrees).Show answer
Answer: -sqrt(3)/2.
Q4In triangle ABC, AB = 7 cm, BC = 9 cm and angle ABC = 65 degrees. Find AC to 3 significant figures.Show answer
Answer: 8.76 cm (cosine rule: AC^2 = 7^2 + 9^2 - 2x7x9xcos(65)).
Q5Solve cos(x) = -0.4 for 0 degrees <= x <= 360 degrees, giving your answers to 1 decimal place.Show answer
Answer: x = 113.6 degrees or x = 246.4 degrees.
Q6Solve 2cos^2(x) - cos(x) - 1 = 0 for 0 degrees <= x <= 360 degrees.Show answer
Answer: x = 0, 120, 240, 360 degrees (from cos(x) = 1 or cos(x) = -1/2).
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
A sector of a circle has radius 10 cm and contains an angle of 2.4 radians. Find the area of the sector, giving your answer to 3 significant figures.
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Solve 3sin(2x - 30 degrees) = 2 for 0 degrees <= x <= 360 degrees, giving your answers to 1 decimal place.
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Show that the equation 3cos^2(x) + sin(x) = 3 can be written as 3sin^2(x) - sin(x) = 0, and hence solve 3cos^2(x) + sin(x) = 3 for 0 degrees <= x <= 360 degrees.
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