Trigonometric Ratios of an Angle

Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.

Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.

A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept out by a particle on its outer edge in $0.15$ seconds.

Evaluate the following expression by replacing the trigonometric ratios with their exact values, without using a calculator:

$\sin^2(30^\circ) + \cos^2(45^\circ) - \tan(60^\circ) \times \sin(60^\circ)$

Calculate the numerical value of the following trigonometric expression corresponding to angles located on the semi-axes:

$3\sin(90^\circ) - 2\cos(180^\circ) + 4\sin(270^\circ) - \cos(360^\circ)$

Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric identities.

Determine which quadrant the angle $\beta$ lies in if both $\sin \beta < 0$ and $\tan \beta > 0$ hold simultaneously.

Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.

In a right-angled triangle, the tangent of one of its acute angles is $\frac{3}{4}$. Determine the exact value of the secant and cosecant of this angle.

Calculate the exact values of $\cot \alpha$ and $\csc \alpha$ given that $\sec \alpha = 3$ and that the sine of the angle is negative.

Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.

Find the exact value of the following expression by first reducing all the angles to the first quadrant:

$\cos(120^\circ) \times \sin(225^\circ) - \tan(330^\circ) \times \cos(210^\circ)$

Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when reducing to that quadrant:

$\frac{\sin(\pi - x) \times \cos\left(\frac{\pi}{2} + x\right) \times \tan(2\pi - x)}{\cos(\pi + x) \times \sin\left(\frac{3\pi}{2} - x\right)}$

A building casts a shadow of $24$ meters when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height of the building and determine its value. If the angle of elevation is halved, does the length of the shadow double? Justify your answer based on the non-linearity of the tangent ratio.

To optimize solar energy collection in winter, a rectangular panel $2$ meters long is raised by resting one of its shorter sides on flat ground, so that the opposite side is at a height of exactly $1.2$ meters. Determine the exact value of the cosine and tangent of the angle of inclination of the panel.

A pendulum of length $80$ cm moves from its vertical equilibrium position to form an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance that the pendulum bob has moved and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.

Prove that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression $A = \frac{n \times R^2}{2} \sin\left(\frac{360^\circ}{n}\right)$. Then, apply the formula to calculate the exact area of a regular octagon inscribed in a circle of radius $10$ cm.

Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:

$\frac{\sin^3 x + \sin x \times \cos^2 x}{\cos x}$

Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are defined:

$\frac{1 - \sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{1 - \sin \alpha} = 2\sec \alpha$

Given the relation $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasons which quadrants the angle $\theta$ can lie in.

Find all angles $\alpha \in [0^\circ, 360^\circ)$ satisfying the following trigonometric equation:

$2\cos^2 \alpha - 3\sin \alpha = 0$

(Hint: express the entire equation in terms of a single trigonometric ratio using the fundamental identity).

A light ray passes from a liquid medium with refractive index $n_1 = 1.33$ to a crystal with index $n_2 = 1.52$. Snell's law states that $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction measured relative to the normal to the surface. If the ray is incident at an angle of $\theta_1 = \frac{\pi}{4}$ rad, determine the exact value of the cosine of the angle of refraction $\theta_2$.

Two circular pulleys of radii $R = 15$ cm and $r = 5$ cm are connected by a taut belt crossed between them. The distance between the centers of the pulleys is $40$ cm. If $\alpha$ is the angle formed by the straight section of the belt with the imaginary line joining the centers, calculate the exact value of $\sin \alpha$, $\cos \alpha$ and the total length of the straight section of the belt between the two points of tangency.

Prove that if $\alpha$ and $\beta$ are two acute complementary angles (that is, $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relation always holds:

$\frac{\tan \alpha + \tan \beta}{\sec \alpha \times \sec \beta} = 1$