Reciprocal trigonometric ratios

Given that $\sin \alpha = \frac{3}{5}$ and that $\alpha$ is an angle in the first quadrant, calculate the exact value of $\csc \alpha$ and $\sec \alpha$.

Consider the acute angle $\alpha = 85^\circ$.

1. a) Without using a calculator to obtain the numerical value, reasonably estimate whether $\sec(85^\circ)$ will be a number less than $1$, close to $1$, or much greater than $1$. Justify your prediction by analysing what happens to the value of $\cos \alpha$ on the unit circle as $\alpha$ approaches $90^\circ$ and taking into account the definition $\sec \alpha = \frac{1}{\cos \alpha}$.
2. b) Knowing that $\cos(85^\circ) \approx 0.0872$, calculate the value of $\sec(85^\circ)$ rounded to two decimal places and check whether your qualitative estimation was correct.
3. c) Based on this behaviour, is it possible to find any real angle $\theta$ such that $\sec \theta = 0.5$? Justify your answer from the range of values that the cosine function can take.

Determine the exact value of $\sec 60^\circ$ and $\cot 45^\circ$ without using a calculator.

Calculate the exact value of $\cot \frac{\pi}{6}$ and $\csc \frac{\pi}{4}$.

Evaluate the following expression giving its exact value, without using decimal approximations: $\sec(120^\circ) + \csc(210^\circ) - \cot(315^\circ)$.

If $\tan \beta = \frac{4}{3}$ and $0 < \beta < \frac{\pi}{2}$, use the fundamental identity $1 + \tan^2 \beta = \sec^2 \beta$ to calculate $\sec \beta$.

Given that $\cos \theta = \frac{1}{3}$, calculate the exact numerical value of the expression $\sec^2 \theta - 1$.

Given that $\sec \alpha = -\frac{13}{5}$ and that $\pi < \alpha < \frac{3\pi}{2}$, calculate the exact value of the remaining trigonometric ratios (sine, cosine, tangent, cosecant and cotangent).

An angle $\gamma$ satisfies $\sec \gamma < 0$ and $\csc \gamma > 0$. Determine, with reasons, which quadrant of the unit circle the angle $\gamma$ lies in.

If $\cot \alpha = 2$ and $\alpha$ belongs to the third quadrant, determine the sign and exact value of $\tan \alpha$ and $\csc \alpha$.

Given that $\csc x = 3$ and that the angle $x$ belongs to the second quadrant, calculate the exact value of the expression $E = \frac{\sec x + \tan x}{\cot x}$.

Simplify the following trigonometric expression as much as possible: $\cos x \times \sec x + \sin x \times \csc x$.

Simplify the following trigonometric expression to obtain a single ratio: $\frac{\sec x}{\csc x} \times \cot^2 x$.

Prove the following trigonometric identity: $\frac{1}{\sec x + \tan x} = \sec x - \tan x$.

Solve the following trigonometric equation, stating all solutions in the interval $[0, 2\pi]$: $2\csc x + 4 = 0$.

Solve the trigonometric equation $\sec^2 x + \tan x = 3$ in the interval $[0^\circ, 360^\circ)$.

Determine all values of $x \in [0, 2\pi)$ satisfying the trigonometric equation $\csc x + \cot x = \sqrt{3}$.

A spotlight illuminates a wall from the ground. The horizontal distance from the spotlight to the base of the wall is $10$ m. The upper light ray forms an angle $\alpha$ with the ground. The length of the light ray from the spotlight to the highest illuminated point on the wall is $L$. Express $L$ using the secant ratio and calculate its exact value if $\cot \alpha = \frac{3}{4}$.

In solar photovoltaic installations in the west of Ireland, the relative air mass ($AM$) models the relative path length that direct radiation travels through the Earth's atmosphere. For zenith angles $\theta$ not too close to the horizon ($0^\circ \le \theta \le 75^\circ$), the approximation $AM = \sec \theta$ is used. The zenith angle $\theta$ and the solar elevation angle above the ground $\beta$ are complementary, that is, $\theta + \beta = 90^\circ$.

In a measurement taken at a weather station in Galway, it is recorded that the cotangent of the solar elevation angle is $\cot \beta = \frac{4}{3}$.

1. a) Prove algebraically, from the definitions of the reciprocal trigonometric functions and the trigonometric ratios of complementary angles, that $\sec \theta = \csc \beta$.
2. b) Without calculating the angle value in degrees and using only the Pythagorean identities from this topic, determine the exact value of the air mass $AM$ at that station.
3. c) Interpret mathematically what happens to the expression $AM = \sec \theta$ when the Sun is located exactly on the horizon (sunset, $\beta = 0^\circ$, which corresponds to $\theta = 90^\circ$). Explain why this model ceases to be applicable at that moment.

In an optics experiment, a laser beam passes through a glass block with flat, parallel faces. The path of the beam inside the glass has length $L = h \times \sec \theta_r$, where $h = 5$ cm is the thickness of the block and $\theta_r$ is the angle of refraction. If the ray is incident from air at an angle $\theta_i = 60^\circ$ and the refractive index of the glass is $n = \sqrt{2}$, calculate the exact length $L$ travelled by the laser inside the block. (Note: Snell's law states that $\sin \theta_i = n \times \sin \theta_r$).