IB Mathematics Analysis and Approaches Topic SL…

Prove the identity
$\frac{1-\tan^2\theta}{1+\tan^2\theta}=\cos(2\theta),$ for all values of $\theta$ for which both sides are defined.

Start from the left-hand side and express everything in terms of $\sin\theta$ and $\cos\theta$ .

Hence, or otherwise, show that $\sin (2 θ) = \frac{2 \tan θ}{1 + \tan^{2} θ}$

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$ .

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$ . The -coordinate of is .

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P$ .

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$ , where is measured in radians.

If $\tan\theta=\dfrac{3}{4}$ , find the exact values of $\sin(2\theta)$ and $\cos(2\theta)$ .

Find all possible values of $\cos\theta$ .

If $P$ lies in the second quadrant, find the exact values of $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ .

Verify that the identity $1+\tan^2(2\theta)=\sec^2(2\theta)$ holds for your values.

Now let $\theta$ vary with $0\le\theta<2\pi$ . Let $Q=(\sqrt{2}\sin\theta-\sqrt{2}\cos\theta)^2$ . the maximum and minimum possible values of , giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$ ) for which the measured quantity $Q$ is greater than $3$ .

If $P$ lies in the fourth quadrant, find the exact values of $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ .

Verify that the identity $1+\tan^2(2\theta)=\sec^2(2\theta)$ holds for your values.

A rotating sensor at $P$ measures a quantity $Q$ given by $(\sqrt{5}\cos\theta-\sqrt{5}\sin\theta)^2$ . the maximum and minimum possible values of $Q$ , giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$ ) for which the measured quantity $Q$ is less than $10$ .

If $P$ lies in the second quadrant, find the exact values of $\sin(2\theta)$ , $\cos(2\theta)$ and $\tan(2\theta)$ .

Verify that the identity $\cot^2(2\theta)+1=\csc^2(2\theta)$ holds for your values.

A rotating beacon at $P$ emits a light beam of intensity $I$ , where $I=(\sin\theta+\cos\theta)^2$ . the maximum and minimum possible intensities, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$ ) for which the intensity is less than $1$ .

IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 3.6—pythagorean Identity, Double Angles Questionbank