Question

Given \(\csc \theta=4,\) use trigonometric identities to find the exact value of (a) \(\sin \theta\) (b) \(\cot ^{2} \theta\) (c) \(\sec \left(90^{\circ}-\theta\right)\) (d) \(\sec ^{2} \theta\)

Short answer

(a) \frac{1}{4}, (b) 15, (c) 4, (d) 16

Step-by-step solution

Find \( \sin \theta \)

Given \(\textrm{csc} \theta = 4\), recall that \(\textrm{csc} \theta = \frac{1}{\textrm{sin} \theta}\). Therefore, \(\textrm{sin} \theta = \frac{1}{4}\).

Find \(\textrm{cot}^{2} \theta\)

Recall the Pythagorean identity \[ \textrm{cot}^{2} \theta = \frac{1}{\textrm{sin}^{2} \theta} - 1 \]. Substitute \(\textrm{sin} \theta = \frac{1}{4}\) to get: \[ \textrm{cot}^{2} \theta = \frac{1}{\frac{1}{16}} - 1 = 16 - 1 = 15 \].

Find \(\textrm{sec} \left( 90^{\circ} - \theta \right) \)

Using the co-function identity, \(\textrm{sec} \left( 90^{\circ} - \theta \right) = \textrm{csc} \theta\). From the given information, \(\textrm{csc} \theta = 4\). Thus, \(\textrm{sec} \left( 90^{\circ} - \theta \right) = 4\).

Find \(\textrm{sec}^{2} \theta \)

Recall that \(\textrm{sec} \theta = \frac{1}{\textrm{cos} \theta}\). Using the identity \[ 1 + \textrm{cot}^{2} \theta = \textrm{csc}^{2} \theta \], we can write \(\textrm{csc}^{2} \theta = 16 (since \textrm{csc} \theta = 4)\). Therefore, \[ 1 + 15 = \textrm{csc}^{2} \theta \] which simplifies to \(\textrm{sec} \theta = \frac{4}{\textrm{cos} \theta}\). Finally, \(\textrm{sec}^{2} \theta = \frac{1}{\textrm{cos}^{2} \theta} = \frac{1}{1 - \textrm{sin}^2 \theta} = \frac{1}{1 - \frac{1}{16}} = 16 \).

Key concepts

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable(s) involved. They are useful for simplifying expressions and solving equations. Some important examples include:

• Pythagorean identities: These relate the squares of sine, cosine, and tangent functions, such as \(\sin^2 \theta + \cos^2 \theta = 1\).
• Reciprocal identities: Involve the relationships between the basic trigonometric functions and their reciprocals, such as \(\csc \theta = \frac{1}{\sin \theta}\).
• Co-function identities: These reveal relationships between trigonometric functions of complementary angles, such as \(\sin(90^{\circ} - \theta) = \cos \theta\).
• Even-odd identities: These describe the symmetry properties of trigonometric functions, like \(\sin(-\theta) = -\sin \theta\) and \(\cos(-\theta) = \cos \theta\).

Knowing these identities helps you solve problems more easily and understand the relationships between different trigonometric functions.

Pythagorean identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It states that for any angle \(\theta\), the sum of the squares of the sine and cosine of that angle is always equal to one: \[ \sin^2 \theta + \cos^2 \theta = 1 \] From this main identity, we can derive other forms, like: \[ 1 + \cot^2 \theta = \csc^2 \theta \] and \[ \tan^2 \theta + 1 = \sec^2 \theta \] These identities are derived by dividing the main identity by either \(\sin^2 \theta\) or \(\cos^2 \theta\), respectively. These forms are very useful in solving trigonometric equations and simplifying expressions. In our exercise, the identity \( 1 + \cot^2 \theta = \csc^2 \theta \) helps us find \(\cot^2 \theta\) knowing \(\csc \theta\).

co-function identity

Co-function identities express the relationship between trigonometric functions of complementary angles. Complementary angles add up to 90 degrees, or \(\pi/2\) radians. These identities include:

• \( \sin(90^{\circ} - \theta) = \cos \theta \)
• \( \cos(90^{\circ} - \theta) = \sin \theta \)
• \( \tan(90^{\circ} - \theta) = \cot \theta \)
• \( \csc(90^{\circ} - \theta) = \sec \theta \)
• \( \sec(90^{\circ} - \theta) = \csc \theta \)
• \( \cot(90^{\circ} - \theta) = \tan \theta \)

In our exercise, we use the co-function identity \( \sec(90^{\circ} - \theta) = \csc \theta \). Given that \(\csc \theta = 4\), we find that \( \sec(90^{\circ} - \theta) = 4 \). This simplifies the problem and allows us to easily find the value of the secant function for the complementary angle.

trigonometric functions

Trigonometric functions are fundamental in mathematics and describe the relationships between the angles and sides of a triangle. The primary trigonometric functions are:

• \(\sin \theta\) - Sine, which represents the ratio of the opposite side to the hypotenuse.
• \(\cos \theta\) - Cosine, which represents the ratio of the adjacent side to the hypotenuse.
• \(\tan \theta\) - Tangent, which is the ratio of the opposite side to the adjacent side.

The reciprocal functions are also essential:

• \( \csc \theta \) - Cosecant, the reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \).
• \( \sec \theta \) - Secant, the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \).
• \( \cot \theta \) - Cotangent, the reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \).

These functions allow us to solve various geometrical problems and are also applicable in different fields such as physics, engineering, and computer science. Understanding the relationships and reciprocal identities between these functions is crucial, as shown in our exercise by using \( \csc \theta \) to find the other function values.