Define \tan\theta for all angles.

\tan\theta=\frac{\sin\theta}{\cos\theta}.

Which ratios are positive in each quadrant?

First: all. Second: \sin. Third: \tan. Fourth: \cos.

State \sin^2\theta+\cos^2\theta.

1 (since x^2+y^2=1 on the unit circle).

Exact values of \sin60^\circ, \cos60^\circ, \tan60^\circ?

\frac{\sqrt3}{2}, \frac12, \sqrt3.

Exact values of \cos150^\circ and \tan150^\circ?

-\frac{\sqrt3}{2} and -\frac{1}{\sqrt3}.

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.

State the cosine rule for a side and for an angle.

a^2=b^2+c^2-2bc\cos A; \cos A=\frac{b^2+c^2-a^2}{2bc}.

State the formula for the area of a triangle.

\frac12ab\sin C, with C between a and b.

What is the ambiguous case?

When the sine rule gives two angles, \theta and 180^\circ-\theta, both of which may form a valid triangle.

When is the second angle 180^\circ-\theta valid in the ambiguous case?

When it and the other known angle still sum to less than 180^\circ.

When can you not use the sine rule directly?

When you have two sides and the included angle, or three sides: use the cosine rule.

Trigonometric ratios, sine and cosine rules and areaEdexcel A-Level Maths: Flashcards

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On the unit circle, what are the coordinates of the point at angle $\theta$?

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On the unit circle, what are the coordinates of the point at angle $\theta$ ?: $(\cos\theta,\sin\theta)$ .
Define $\tan\theta$ for all angles.: $\tan\theta=\frac{\sin\theta}{\cos\theta}$ .
Which ratios are positive in each quadrant?: First: all. Second: $\sin$ . Third: $\tan$ . Fourth: $\cos$ .
State $\sin(180^\circ-\theta)$ and $\cos(180^\circ-\theta)$ .: $\sin\theta$ and $-\cos\theta$ .
State $\sin^2\theta+\cos^2\theta$ .: $1$ (since $x^2+y^2=1$ on the unit circle).
Exact values of $\sin60^\circ$ , $\cos60^\circ$ , $\tan60^\circ$ ?: $\frac{\sqrt3}{2}$ , $\frac12$ , $\sqrt3$ .
Exact values of $\cos150^\circ$ and $\tan150^\circ$ ?: $-\frac{\sqrt3}{2}$ and $-\frac{1}{\sqrt3}$ .
State the sine rule.: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ .
State the cosine rule for a side and for an angle.: $a^2=b^2+c^2-2bc\cos A$ ; $\cos A=\frac{b^2+c^2-a^2}{2bc}$ .
State the formula for the area of a triangle.: $\frac12ab\sin C$ , with $C$ between $a$ and $b$ .
What is the ambiguous case?: When the sine rule gives two angles, $\theta$ and $180^\circ-\theta$ , both of which may form a valid triangle.
When is the second angle $180^\circ-\theta$ valid in the ambiguous case?: When it and the other known angle still sum to less than $180^\circ$ .
When can you not use the sine rule directly?: When you have two sides and the included angle, or three sides: use the cosine rule.

Exam questions on Trigonometric ratios, sine and cosine rules and area

A point $P$ lies on the unit circle centred at the origin $O$ . The angle from the positive $x$ -axis to $OP$ , measured anticlockwise, is $150^\circ$ .
The point $Q$ on the unit circle corresponds to an angle of $210^\circ$ . Write down the exact coordinates of $Q$ .2 marks
In triangle $ABC$ , $AB=6$ cm, $BC=10$ cm and $A\hat{B}C=60^\circ$ .
Find the size of angle $B\hat{A}C$ , giving your answer to one decimal place.2 marks
The angle $\theta$ satisfies $\sin\theta=\frac35$ and $90^\circ<\theta<180^\circ$ . A triangle has two sides of lengths $10$ cm and $13$ cm with included angle $\theta$ .
Find the exact values of $\cos\theta$ and $\tan\theta$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).