Exercises: Trigonometric ratios of an angle

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 degrees.

Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right triangle knowing that its legs measure $5$ inches and $12$ inches.

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 the fundamental trigonometric identities.

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

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

Determine in which quadrant the angle $\beta$ is located if it is simultaneously true that $\sin \beta < 0$ and $\tan \beta > 0$.

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 its cotangent.

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

A building casts a shadow of $24$ feet 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 nonlinearity of the tangent ratio.

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

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

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 by a particle on its outer edge in $0.15$ seconds.

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

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

A pendulum of length $80$ inches is displaced from its vertical equilibrium position until it forms an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance the mass of the pendulum has been displaced and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.

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

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

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

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

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

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

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 this quadrant:

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

Find all angles $\alpha \in [0^\circ, 360^\circ)$ that satisfy 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 ray of light 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 with respect 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 $R = 15$ inches and $r = 5$ inches are connected by a tight belt that crosses between them. The distance between the centers of the pulleys is $40$ inches. 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 complementary acute angles (i.e., $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relationship is always verified:

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