Introduction to Trigonometry

In a right triangle $ABC$ with $\angle B = 90^\circ$, if $\cos A = 0.6$, what is the length of $AB$ if $AC = 10$?

In $\triangle ABC$, right-angled at $B$, if $AB = BC$, find the value of $\sec A$.

In a right triangle $ABC$ ($\angle B = 90^\circ$), if $AC = 5$ and $BC = 3$, what is $\csc A$?

In $\triangle ABC$ right-angled at $B$, if $\tan A = \sqrt{3}$, find the value of $\cos A$.

If $\sin \theta = \frac{p}{q}$, then $\cos \theta$ is equal to:

If $\tan \theta = \frac{3}{4}$, then what is the value of $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta}$?

In a right triangle $ABC$, right-angled at $B$, what is $\sin A + \cos C$ if $AB=4, BC=3, AC=5$?

A wooden plank of length $13$ units is propped against a wall. If the height of the wall reached by the plank is $12$ units, let $\phi$ be the angle between the plank and the ground. Calculate $\sqrt{\frac{1 - \sin \phi}{1 + \sin \phi}}$.

In the given figure, if $AD = 4$ units, $DB = 3$ units, and $\angle ADE = \angle ABC = 90^\circ$, find the value of $\sin A$ and use it to show that the ratio $DE/BC$ depends only on $AD/AB$. Calculate $\sin A$.

A park has a slide shaped like a right triangle. If the height of the slide is $3$ m and the base is $4$ m, let $\theta$ be the angle of the slide with the base. Find the value of $5 \sin \theta - 3 \tan \theta$.

The graph below shows the function $y = \sin(x)$ for angles between $0^\circ$ and $90^\circ$. Based on the standard trigonometric values, what is the value of $y$ when $x = 45^\circ$?

In the given right-angled triangle $ABC$, the angle $\angle ABC$ is $90^\circ$ and $\angle BAC = 30^\circ$. If the length of the hypotenuse $AC$ is $8$ units, find the value of $BC$ by evaluating $\sin 30^\circ$.

Find the value of $4 \sin^2 60^\circ + 3 \tan^2 30^\circ - 8 \sin 45^\circ \cos 45^\circ$.

What is the value of $(\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ)$?

If $\sin(A + B) = 1$ and $\cos(A - B) = \frac{\sqrt{3}}{2}$, where $0 < A + B \le 90^\circ$, find $A$.

Calculate: $\frac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}$.

Find the value of $x$ in the equation: $\sqrt{3} \tan 2x = \cos 60^\circ + \sin 45^\circ \cos 45^\circ$.

Find the value of $x$ if $\sqrt{3} \tan(2x) = \cos 60^\circ + \sin 45^\circ \cos 45^\circ$.

A flagstaff of height $h$ stands on the top of a tower. From a point on the ground, the angle of elevation of the bottom of the flagstaff is $30^\circ$ and that of the top is $45^\circ$. If the tower's height is $50$ m, find the height of the flagstaff.

If $\sin(A-B) = 1/2$ and $\cos(A+B) = 1/2$, where $0 < A+B \le 90^\circ$ and $A > B$, find the values of $A$ and $B$.

In the given right-angled triangle $ABC$, the angle at $B$ is $90^\circ$ and $\angle BAC = \theta$. Given that $\sin \theta = \frac{3}{5}$, what is the value of $\cos \theta$?

Find the value of $\frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}}$.

If $\sin A = \frac{1}{2}$, find the value of $3 \cos A - 4 \cos^3 A$.

The value of $(\sin 30^{\circ} + \cos 30^{\circ}) - (\sin 60^{\circ} + \cos 60^{\circ})$ is:

Given the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, evaluate the following expression for any acute angle $\theta$:

$E = (\sec^2 \theta - 1) \cot^2 \theta$

If $A + B = 90^{\circ}$, then $\sqrt{\frac{\tan A \tan B + \tan A \cot B}{\sin A \sec B} - \frac{\sin^2 B}{\cos^2 A}}$ simplifies to:

What is the value of $\sin^2 30^{\circ} + \sin^2 45^{\circ} + \sin^2 60^{\circ}$?

If $\sin \theta + \cos \theta = \sqrt{3}$, find the value of $\tan \theta + \cot \theta$ to calculate the torque in a rotating mechanical system.

A bridge truss is designed such that $\frac{1 + \tan^2 A}{1 + \cot^2 A}$ defines the ratio of two supporting forces. Simplify this expression.

In a geometry task, a student is asked to simplify the expression $(\sec A + \tan A)(1 - \sin A)$. What is the resulting trigonometric ratio?

Given $\triangle ABC$ is right-angled at $C$, what is the value of $\cos (A+B)$?

In a right triangle $ABC$, right-angled at $B$, if $BC = 7$ and $AC = 25$, find $\tan C$.

Evaluate: $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$.

In triangle $ABC$ right-angled at $B$, $AB = 5$ cm and $\angle ACB = 30^\circ$. Find the length of the side $BC$.

In the given right-angled triangle $ABC$, right-angled at $B$, the side $AB$ measures $7$ units and $BC$ measures $24$ units. If $\angle ACB = \theta$, what is the value of $\sin \theta + \cos \theta$?

If $\tan \theta = \frac{4}{3}$, then the value of $\sqrt{\frac{1-\sin \theta}{1+\sin \theta}}$ is:

If $\sin \theta - \cos \theta = 0$, then find the value of $\sin^4 \theta + \cos^4 \theta$.

If $\sec \theta = \frac{13}{5}$, evaluate the expression $\frac{2 \sin \theta - 3 \cos \theta}{4 \sin \theta - 9 \cos \theta}$.

In $\triangle ABC$, $\angle B = 90^{\circ}$ and $\tan A = \frac{1}{\sqrt{3}}$. Calculate the value of $\sin A \cos C + \cos A \sin C$.

A slide in a park is $10$ m long and makes an angle $\theta$ with the ground. If $\sin \theta = 0.8$, how high is the top of the slide from the ground?

Given the right triangle $PQR$ shown below, where $\angle Q = 90^\circ$ and $PR = 13$ cm, $QR = 5$ cm. Determine the value of $\cos P$.

In the given right-angled triangle $ABC$, right-angled at $B$, the lengths of sides $AB$ and $BC$ are $3$ units and $4$ units respectively. Find the value of $\sin A$.

Which of the following is equal to $\sec^2 \theta - 1$?

If $\sin \theta = \cos \theta$, then the value of $\theta$ (where $0^\circ \le \theta \le 90^\circ$) is:

In triangle $XYZ$ right-angled at $Y$, if $\tan X = \frac{1}{\sqrt{3}}$, calculate the value of $\sin X \cos Z + \cos X \sin Z$.

The value of the expression $\frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ}$ is:

If $\cos(40^\circ + x) = \sin 30^\circ$, find the value of $x$.

In a right-angled triangle $ABC$, $\angle B = 90^\circ$. If $AC - AB = 1$ and $BC = 5$, find the value of $\sec A + \tan A$.

A flagpole is $h$ meters high. At a point $P$ on the ground, the angle of elevation to the top of the pole is $\beta$. If $\cos \beta = \frac{21}{29}$, find the value of $\frac{h}{d}$, where $d$ is the distance from $P$ to the base of the pole.

The coordinates of a point $P$ on a circle centered at the origin $O(0,0)$ are $(x, y)$. If the radius is $r$ and the line $OP$ makes an angle $\alpha$ with the positive x-axis, which expression represents $\tan \alpha + \cot \alpha$?

In the given right-angled triangle $\triangle ABC$, the angle $\angle B$ is $90^\circ$ and $\angle C = 30^\circ$. If the length of the hypotenuse $AC$ is $12$ cm, find the length of the side $AB$.

The value of $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$ is equal to:

Which trigonometric ratio of $45^\circ$ is equal to $1$?

Identify the value of $x$ if $2 \sin 2x = \sqrt{3}$ for $0^\circ \le 2x \le 90^\circ$.

In $\triangle ABC$, $\angle B = 90^\circ$, $BC = 15$, and $\angle C = 60^\circ$. Find the length of $AC$.

If $x \sin 45^\circ \cos 45^\circ \tan 60^\circ = \tan^2 45^\circ - \cos 60^\circ$, find $x$.

What is the value of $\frac{1 - \cos^2 30^\circ}{1 - \sin^2 30^\circ}$?

An observer $1.5$ meters tall is $28.5$ meters away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^{\circ}$. What is the height of the chimney?

A ladder $15$ meters long just reaches the top of a vertical wall. If the ladder makes an angle of $60^{\circ}$ with the wall, find the height of the wall.

Two poles of equal heights are standing opposite each other on either side of a road, which is $80$ meters wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^{\circ}$ and $30^{\circ}$, respectively. Find the height of the poles.

What is $(\sec^{2} \theta - 1) \cot^{2} \theta$?

Simplify: $\sin A \cdot \csc A - \cos A \cdot \sec A$.

Which of these is the correct identity for $\cot^{2} A$?

If $\sin \theta = \frac{1}{2}$, then $\sin^{2} \theta + \cos^{2} \theta$ is:

If $\sin \theta + \sin^{2} \theta = 1$, then find the value of $\cos^{2} \theta + \cos^{4} \theta$.

What is the value of $\frac{\cos^{3} A + \sin^{3} A}{\cos A + \sin A} + \frac{\cos^{3} A - \sin^{3} A}{\cos A - \sin A}$?

Evaluate: $(\csc \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)$.

In a geometry problem, a circle is inscribed in a square. A calculation results in the expression $\frac{1}{1 + \sin^2 \theta} + \frac{1}{1 + \csc^2 \theta}$. What is the simplified value of this expression?

An environmental scientist models the tide using the expression $T = \sin^6 \theta + \cos^6 \theta$. To predict the peak, she needs an alternative form involving $\sin^2 \theta \cos^2 \theta$. What is the correct identity?

A computer graphics algorithm calculates shading using the expression $(\sec^2 \theta + \csc^2 \theta)$. For optimization, the programmer wants to write this as a product of two functions. Which of the following is correct?