# Trigonometry

## Self-Test:

 1)       $\left[ \sin\left(\frac{\pi}{4}\right)+\cos\left(\frac{5\pi}{6}\right) \right] \times \left[\frac{\arctan(1)}{\pi}\right]=$ $\frac{\sqrt{2}-\sqrt{3}}{8}$ $\frac{2(\sqrt{2}-\sqrt{3})}{\pi^2}$ $\frac{\sqrt{2}-1}{8}$ $\frac{\sqrt{2}+\sqrt{3}}{\pi}$ $\frac{\sqrt{2}-\sqrt{3}}{2}$

Hint Use the unit circle and the $(30-60-90)$ and $(45-45-90)$ triangles to find the trigonometric values.

 2)       As the angle $\theta$ increases towards $\frac{\pi}{2}$, the value of $\tan{\theta}$ goes to $0$ $+\infty$ $-\infty$ $1$ $-1$

Hint Use the graph of $y=\tan{\theta}$ or consider the ratio $\tan{\theta}=\frac{\sin{\theta}}{\cos{\theta}}$.

 3)       Let $x=3\cos{\theta}$ and $y=3\sin{\theta}$. Then $\sqrt{x^2+y^2}=$ $\sqrt{3}$ $\sqrt{3\theta}$ $\sqrt{3}(cos{\theta}+\sin{\theta})$ $3$ $9$

Hint One of the trigonometric identities states that $\sin^2{\theta}+\cos^2{\theta}=1$.

 4)       $\frac{\cos{\theta}}{(1-\cos{\theta})(1+\cos{\theta})}=$ $\frac{1}{1+\cos{\theta}}$ $\csc{\theta}\tan{\theta}$ $\csc{\theta}\cot{\theta}$ $\sec{\theta}\cot{\theta}$ $\frac{1+\cos{\theta}}{\sin^2{\theta}}$

Hint The difference of squares formula tells us that $(A+B)(A-B)=A^2-B^2$ and we have the trigonometric identity $\sin^2{\theta}+\cos^2{\theta}=1$.

 5)       Let the point $P=\left(\cos\left(\frac{\pi}{12}\right),\sin\left(\frac{\pi}{12}\right)\right)$. Then $P$ is located: strictly inside the unit circle. on the unit circle. strictly outside the unit circle, in the upper half plane. strictly outside the unit circle, to the left of the $y$-axis. strictly outside the unit circle, in the lower half plane.

Hint We have the trigonometric identity $\sin^2{\theta}+\cos^2{\theta}=1$ for any angle $\theta$.

 6)       Which of the following statements is true? $3 \text{ radians} < 50^{\circ}$ $1 \text{ radian} \neq 1^{\circ}$ $2\pi \text{ radians} < 360^{\circ}$ $\frac{\pi}{2} \text{ radians} \neq 90^{\circ}$ $1.5 \text{ radians} > 180^{\circ}$

Hint We have that $1 \text{ radian}=\frac{360^{\circ}}{2\pi}$

 7)       A path that travels $5$ radians around the circumference of a circle covers less than half of the circle. more than half of the circle, but less than $1$ full rotation. more than a full rotation. more than 1 full rotation, but less than 2 full rotations. exactly 5 full rotations.

Hint We know that $2\pi \text{ radians}=360^{\circ}$ which takes us around the circle once.

 8)       Triangle $ABC$ has side lengths $a=10$ and $b=16$. Across from side $a$, angle $A$ is $30$ degrees. The correct values (up to one decimal place) of the other two angles are: $B=53.1^{\circ}, C=96.9^{\circ}$ or $B=60^{\circ}, C=90^{\circ}$ $B=126.9^{\circ}, C=23.1^{\circ}$ or $B=60^{\circ}, C=90^{\circ}$ $B=53.1^{\circ}, C=96.9^{\circ}$ or $B=36.9^{\circ}, C=113.1^{\circ}$ $B=53.1^{\circ}, C=96.9^{\circ}$ or $B=126.9^{\circ}, C=23.1^{\circ}$ $B=53.1^{\circ}, C=96.9^{\circ}$ only.

Hint 1 The Sine Law states that for a triangle $ABC$, $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$.
Hint 2 We have the identity $\sin(\pi-\theta)=\sin(\theta)$.

 9)       A building casts a shadow with a length of $2$ metres when the sun is at an angle of elevation of $35$ degrees from the ground. The height of the building is equal to: $\cos(55^{\circ})$ $\sin(55^{\circ})$ $\tan(55^{\circ})$ $2\cos(35^{\circ})$ $2\tan(35^{\circ})$

Hint In a right-angle triangle, the trigonometric functions represent ratios of the side lengths.

 10)       After walking for $3$ metres along the shore of a river and then swimming $4$ metres perpendicular to the shore, you are now $x$ metres from your starting position, having formed a triangle with these angles (rounded to one decimal place): $x=5$, with angles $30^{\circ}, 60^{\circ},90^{\circ}$ $x=25$, with angles $30^{\circ}, 60^{\circ},90^{\circ}$ $x=5$, with angles $36.9^{\circ}, 53.1^{\circ},90^{\circ}$ $x=25$, with angles $36.9^{\circ}, 53.1^{\circ},90^{\circ}$ $x=2$, with angles $38^{\circ}, 52^{\circ},90^{\circ}$

Hint Use the Pythagorean theorem to find the third side in the right-angle triangle. Use the inverse trigonometric functions to find the angles.