What does the symbol \equiv mean?

The two sides are equal for every valid value of the variable (an identity).

\sin\theta=\frac35, \theta acute: find \cos\theta and \tan\theta.

\cos\theta=\frac45, \tan\theta=\frac34.

How do you choose the sign of \cos\theta when you take a square root?

Use the quadrant: cosine is positive in the 1st and 4th quadrants.

How do you prove an identity?

Work on one side only, with every step shown, until it matches the other side.

Why is \tan\theta undefined at 90^\circ?

Because \cos90^\circ=0 and you cannot divide by zero.

Trigonometric identitiesEdexcel International A Level Maths: Flashcards

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Question

State the identity linking $\tan\theta$ to $\sin\theta$ and $\cos\theta$.

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State the identity linking $\tan\theta$ to $\sin\theta$ and $\cos\theta$ .: $\tan\theta=\frac{\sin\theta}{\cos\theta}$ .
State the main Pythagorean trigonometric identity.: $\sin^2\theta+\cos^2\theta=1$ .
Rearrange the identity to give $\sin^2\theta$ .: $\sin^2\theta=1-\cos^2\theta$ .
Rearrange the identity to give $\cos^2\theta$ .: $\cos^2\theta=1-\sin^2\theta$ .
What does the symbol $\equiv$ mean?: The two sides are equal for every valid value of the variable (an identity).
Simplify $\tan\theta\cos\theta$ .: $\sin\theta$ .
Simplify $\frac{\sin^2\theta}{1-\sin^2\theta}$ .: $\tan^2\theta$ .
Expand $(\sin\theta+\cos\theta)^2$ and simplify.: $1+2\sin\theta\cos\theta$ .
$\sin\theta=\frac35$ , $\theta$ acute: find $\cos\theta$ and $\tan\theta$ .: $\cos\theta=\frac45$ , $\tan\theta=\frac34$ .
How do you choose the sign of $\cos\theta$ when you take a square root?: Use the quadrant: cosine is positive in the 1st and 4th quadrants.
In which quadrants is $\tan\theta$ positive?: The 1st and 3rd.
How do you prove an identity?: Work on one side only, with every step shown, until it matches the other side.
Why is $\tan\theta$ undefined at $90^\circ$ ?: Because $\cos90^\circ=0$ and you cannot divide by zero.

Exam questions on Trigonometric identities

$\theta$ is an obtuse angle with $\sin\theta=\frac{8}{17}$ .
Find the exact value of $\sin\theta\cos\theta$ .2 marks
$\cos x=-\frac23$ and $180^\circ<x<270^\circ$ .
Find the exact value of $\tan^2x-\sin^2x$ .2 marks
In this question, $\theta$ is any angle for which the expressions are defined.
Show that $(\sin\theta+\cos\theta)^2\equiv1+2\sin\theta\cos\theta$ .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).