Trigonometry

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.