Trigonometric identities and their uses · Year 12
In Trigonometric identities and their uses (Year 12) you practise one skill: Simplifying with the basic identities. A trigonometric identity holds for every angle. Besides the identity that the squares of the sine and the cosine add up to one, there are identities for the sine and the cosine of the sum and of the difference of two angles, and the double angle identities follow from them: the sine of a double angle is twice the sine times the cosine, and the cosine of a double angle is the square of the cosine minus the square of the sine. There are also identities that turn a sum of two sines or of two cosines into a product. The identities simplify expressions, solve equations and give angles and lengths in plane figures. In solids, such as a pyramid and a cone, this is only a first look, with no solid geometry theorems: find a right triangle inside the solid and solve it.
A model proof: the cosine of the difference of two angles. On the unit circle mark two points: the point of the angle alpha, whose coordinates are the cosine and the sine of alpha, and the point of the angle beta, whose coordinates are the cosine and the sine of beta. The angle between them at the centre is alpha minus beta. Work out the square of the distance between the two points in two ways. From the coordinates, it is the square of the difference of the cosines plus the square of the difference of the sines. Open the brackets: the square of the sine of an angle plus the square of its cosine is one, so this is two minus twice the sum of the cosine of alpha times the cosine of beta and the sine of alpha times the sine of beta. By the cosine rule in the triangle made by the centre and the two points, whose two sides from the centre are radii of length one, the square of the distance is two minus twice the cosine of alpha minus beta. The two ways agree, so the cosine of alpha minus beta equals the cosine of alpha times the cosine of beta, plus the sine of alpha times the sine of beta. From the identity we proved: the cosine of a sum. Put minus beta in place of beta. The cosine of minus beta is the cosine of beta, and the sine of minus beta is minus the sine of beta. So the cosine of alpha plus beta equals the cosine of alpha times the cosine of beta, minus the sine of alpha times the sine of beta. From the identity we proved: the sine of a sum. The sine of an angle is the cosine of ninety degrees minus the angle, so the sine of alpha plus beta is the cosine of ninety degrees minus alpha, minus beta. By the identity we proved, that is the cosine of ninety degrees minus alpha times the cosine of beta, plus the sine of ninety degrees minus alpha times the sine of beta, that is the sine of alpha times the cosine of beta, plus the cosine of alpha times the sine of beta. From the identity we proved: the sine of a difference. Put minus beta in place of beta in the identity for the sine of a sum: the sine of alpha minus beta equals the sine of alpha times the cosine of beta, minus the cosine of alpha times the sine of beta. The double angle identities follow from the identities for a sum: put beta equal to alpha.
Tip: Before dividing an equation by a sine or a cosine, check whether it can be zero. Take out the common factor instead, and no solution is lost.
What you learn
Simplifying with the basic identities
For example: Simplify the expression (1 − cos²x) ÷ sin x .
- cos x
- tan x
- 1
- sin x
Answer and solution
The answer: sin x
By sin²x + cos²x = 1 , the numerator 1 − cos²x equals sin²x . Cancel sin x and sin x is left.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).