Exercises: 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 $\sec \alpha$.

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

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 $\sec \beta$.

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

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.

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

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

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

Lucía has to calculate the value of $\sec \alpha$ knowing that $\tan \alpha = -\frac{4}{3}$ and that $\alpha$ is a second-quadrant angle ($90^\circ < \alpha < 180^\circ$). In her exam she writes the following procedure:

At the end she adds: "Since the secant comes from solving a square root of a sum of squared terms, its result is always positive".

1. a) Reasonably explain the two conceptual errors Lucía makes (one in the algebraic solving and another in her statement about the sign of the secant), relying on the relationship between the secant and the cosine and on the position of the angle on the unit circle.
2. b) Correct the calculation and find the exact value of $\sec \alpha$ and $\csc \alpha$.

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.

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).

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

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

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 $\cot \alpha = \frac{3}{4}$.

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

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

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

Prove that for any angle $\alpha$ where the ratios are defined, the following equality holds: $\cot^2 \alpha - \cos^2 \alpha = \cot^2 \alpha \cdot \cos^2 \alpha$.

In solar photovoltaic installations in Extremadura, 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 Cáceres, 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.

To simplify the trigonometric expression $E(x) = \frac{\sec^2 x + \csc^2 x}{\sec^2 x \cdot \csc^2 x}$ at all points where it is defined, two 1st-year Bachillerato students propose different strategies:

• Clara's method: Express all ratios in terms of $\sin x$ and $\cos x$ from the beginning, carry out the addition of fractions in the numerator, and then divide the resulting fraction by the denominator.

• Marcos's method: Decompose the original algebraic fraction into a sum of two fractions with the same common denominator, $\frac{\sec^2 x}{\sec^2 x \cdot \csc^2 x} + \frac{\csc^2 x}{\sec^2 x \cdot \csc^2 x}$, simplify each term, and apply the fundamental identities.

1. a) Apply both methods step by step and verify that both lead to a constant numerical expression.
2. b) Compare both approaches and justify which one is more efficient, pointing out the algebraic advantages it has over the other.
3. c) Determine which angles in the interval $[0^\circ, 360^\circ]$ must be excluded from the domain of $E(x)$ because they make any of the original denominators of the involved reciprocal functions equal to zero.

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

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

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 $L = h \cdot \sec \theta_r$, 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 $\sin \theta_i = n \cdot \sin \theta_r$).

Simplify the following trigonometric expression and express it in terms of a single trigonometric ratio or numerical constant: $\frac{\sec^4 x - 1}{\tan^2 x} - \tan^2 x$.

Solve the following system of trigonometric equations for values of $x, y \in [0, \pi]$:

An L-shaped hallway has two sections of widths $a = 2$ m and $b = 3$ m that intersect at a right angle. A rigid beam of length $L$ gets stuck while attempting to turn the corner, touching both outer walls and the inner corner. The length of the beam as a function of the angle $\theta$ it makes with one of the outer walls can be modeled as $L(\theta) = a \sec \theta + b \csc \theta$. Calculate the exact value of $L(\theta)$ when $\cot \theta = \sqrt[3]{\frac{a}{b}}$.

Prove the following trigonometric identity, showing the necessary algebraic steps: $\frac{\csc \alpha + \cot \alpha}{\sec \alpha + \tan \alpha} = \frac{\sec \alpha - \tan \alpha}{\csc \alpha - \cot \alpha}$.

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

You want to construct a quadratic equation of the form $A x^2 + B x + C = 0$, with integer coefficients such that $A > 0$ and $\gcd(A, |B|, |C|) = 1$, whose real solutions are exactly $x_1 = \sec \alpha$ and $x_2 = \csc \alpha$.

It is known that $\alpha$ is a first-quadrant angle ($0^\circ < \alpha < 90^\circ$) that satisfies the following condition:

1. a) Without calculating the angle $\alpha$, square the equation above and apply the fundamental Pythagorean identity to deduce the exact value of the product $\sin \alpha \cdot \cos \alpha$.
2. b) Using this product and the definitions of the reciprocal functions, calculate the sum of the roots $S = \sec \alpha + \csc \alpha$ and the product of the roots $P = \sec \alpha \cdot \csc \alpha$ in the form of irreducible fractions.
3. c) From $S$ and $P$, write the equation $A x^2 + B x + C = 0$ with the requested integer coefficients.
4. d) Solve the resulting quadratic equation to find the numerical values of $\sec \alpha$ and $\csc \alpha$, and justify why it was essential for both solutions to be strictly greater than $1$ considering that $\alpha$ belongs to the first quadrant.

David has to solve the trigonometric equation $\sec x - \tan x = \cos x$ in the interval $[0, 2\pi)$ on an exam. His working is as follows:

1. He multiplies both sides by $\cos x$: $\cos x \cdot \sec x - \cos x \cdot \tan x = \cos^2 x \implies 1 - \sin x = \cos^2 x$.

2. He substitutes $\cos^2 x = 1 - \sin^2 x$, obtaining $1 - \sin x = 1 - \sin^2 x$.

3. He cancels the $1$ and factorises: $\sin^2 x - \sin x = 0 \implies \sin x (\sin x - 1) = 0$.

4. He concludes that the solutions are: if $\sin x = 0 \implies x = 0$ or $x = \pi$; if $\sin x = 1 \implies x = \frac{\pi}{2}$. David writes as the final solution: $S = \left\{0, \pi, \frac{\pi}{2}\right\}$.

1. a) Identify which of the values obtained by David does not belong to the domain of definition of the original equation and explain the analytical and geometric reason why $\sec x$ and $\tan x$ do not exist at that value.
2. b) Explain what algebraic error was made in step 1 that caused this extraneous or false solution to appear.
3. c) Check whether the other two values ($x = 0$ and $x = \pi$) are indeed solutions by substituting them into the original equation, and write down the rigorous solution set of the problem.