Trigonometry: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)

Related topics:
Trigonometric identities
Trigonometric equations

Problem 1
$\text{Sin} \theta ,\cos \theta ,\tan \theta ,\cot \theta$ are respectively:

Problem 2
$\text{Sin} \theta ,\cos \theta ,\tan \theta ,\cot \theta$ are respectively:

Problem 3
$c=3\qquad a=\sqrt{5}$ Find b = ? and $\sin \theta = ?$

Problem 4
$\theta =30^{\circ }\qquad a=2$
What are the lengths of $b$ and $c$?

Problem 5
$\theta =30^{\circ }\qquad a=2$
What are the lengths of $a$ and $c$?

Problem 6
$\sin \theta=?$

Problem 7
A kite is stuck in the branches of a tree. If the kite's $90$ feet string makes an angle of $22^{\circ }$ to the ground, calculate the distance between the kite and the ground.
Problem 8
If you know that $\sin \theta =\frac{2}{\sqrt{13}}$ and $\cos \theta =\frac{3}{\sqrt{13}}$, calculate $\tan \theta$ and $\cot \theta$.
Problem 9
A carpenter cuts the end of a four inch board, forming a bevel of $25^\circ$ with respect to the vertical, starting at a point $1\frac{1}{2}$ inches from the end of the board. Calculate the lengths of the diagonal cut and the remaining side.
Problem 10
Angle $\theta$ is acute. If $\sin \theta =\frac{1}{\sqrt{65}},\tan \theta =\frac{1}{8}$ then

Problem 11
If $\cos 75^{\circ }=\frac{1}{4}\left( \sqrt{6}-\sqrt{2}\right)$, find the exact value of $\sin 15^{\circ }$.
Problem 12
$\frac{\sin 2x}{1+\cos 2x}=$
Problem 13
$\frac{1-\sin ^{4}\alpha }{\cos ^{2}\alpha }=$
Problem 14
$\frac{2\sin \alpha }{\tan 2\alpha }=$
Problem 15
$\frac{1-\sin \alpha }{\cos \alpha }=$
Problem 16
$\cos ^{2}\alpha =$
Problem 17
As seen in figure , two tracking stations $S_{1}$ and $S_{2}$ spot a weather balloon between them, with respective elevation angles $\alpha $ and $\beta$.
Express the height $h$ of the balloon as a function of $\alpha $ and $\beta$, and the distance $c$ between the tracking stations.
Suppose the stations and the balloon are on the same vertical plane.
Problem 18
A surveyor uses an instrument called a theodolite to measure the angle of elevation between the ground level and the top of a mountain. At one point, a forty degree elevation angle is measured. Half a kilometer further from the base of the mountain, the angle of elevation measured is $37^{\circ}$. How high is the mountain?

Problem 19
Determine the exact value of $\cot \left( \frac{3\pi }{8}\right)$ if $\tan \left( \frac{\pi }{8}\right) =\sqrt{2}-1$
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