Question

Without using a calculator, evaluate or simplify the following expressions. $$\csc ^{-1}(\sec 2)$$

Short answer

Question: Evaluate the given expression without using a calculator: \(\csc^{-1}(\sec 2)\). Answer: \(90^\circ - 2\)

Step-by-step solution

Rewrite the trigonometric functions

Since the given expression includes the inverse function notation, let's first rewrite it as an angle: $$\theta = \csc^{-1}(\sec 2)$$ This means the cosecant of angle \(\theta\) is equal to the secant of \(2\) degrees, in other words: $$\csc{\theta} = \sec 2$$

Write the trigonometric functions in terms of sine and cosine

To simplify further, we can rewrite the cosecant and secant functions in terms of sine and cosine, since they are reciprocals of sine and cosine respectively: $$\frac{1}{\sin{\theta}} = \frac{1}{\cos 2}$$

Use the cofunction identity

Since we are dealing with the opposite function of sine (cosecant) and cosine (secant), we can use the cofunction identity to express one of the functions in terms of the other: $$\cos x = \sin (90^\circ - x)$$ Let's apply the cofunction identity to our given expression: $$\frac{1}{\sin{\theta}} = \frac{1}{\sin (90^\circ - 2)}$$

Simplify the expression

Now that we have expressed both functions in terms of sine, we can simplify the expression as follows: $$\sin{\theta} = \sin (90^\circ - 2)$$ Therefore, the angle \(\theta\) is equivalent to \(90^\circ - 2\).

Find the final solution

We initially expressed the given expression as an angle, so now we can write the final solution back in terms of the inverse function notation: $$\csc^{-1}(\sec 2) = \theta = 90^\circ - 2$$

Key concepts

Cofunction Identity

The cofunction identity is an important concept in trigonometry that relates different trigonometric functions. It states that the sine of an angle is equal to the cosine of its complement, meaning:

• \( \cos x = \sin (90^\circ - x) \)
• Similarly, \( \sin x = \cos (90^\circ - x) \)

This identity is particularly useful when dealing with problems involving angles in a right triangle or when simplifying expressions involving complementary angles.
In the original exercise, the concept is used to transform an expression involving cosecant and secant into a more manageable form. Since cosecant and secant are the reciprocals of sine and cosine, respectively, the identity helps by enabling the simplification:

• We express cosecant in terms of sine and use the identity to match it with cosine.
• By rewriting \( \csc \theta = \sec 2 \) into \( \frac{1}{\sin \theta} = \frac{1}{\sin (90^\circ - 2)} \), we can simplify and find \( \theta \) as a function of angle expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables, ensuring consistency across various trigonometric applications.
They simplify the process of solving equations or verifying the equality of two expressions by providing established relationships. Fundamental trigonometric identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal Identities: \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \)
• Cofunction Identities like the one used here: \( \cos x = \sin (90^\circ - x) \)

In the given exercise, the cofunction identity and reciprocal identities are key to transforming and simplifying the expression \( \csc^{-1}(\sec 2) \).
By rewriting the expression using trigonometric identities, it greatly simplifies solving or evaluating complex trigonometric expressions.

Trigonometric Function Simplification

Simplifying trigonometric functions often involves reducing complex expressions into simpler terms using established identities and properties. The main goal is to transform a complicated expression into an equivalent but reduced form.
In the given problem, simplification involves several steps and uses multiple identities:

• First, rewrite trigonometric functions in terms of sine and cosine using reciprocal identities.
• Apply the cofunction identity to facilitate rearrangement and reduction of terms.
• Use logical reasoning to equate both sides of the simplified equation to deduce the angle or expression result.

By expressing \( \frac{1}{\sin \theta} = \frac{1}{\sin (90^\circ - 2)} \), simplification leads to finding \( \theta = 90^\circ - 2 \), thus resolving the inverse trigonometric function back to its standard form.
Trigonometric function simplification is a crucial step in solving trigonometric equations and understanding trigonometric behavior across various mathematical contexts.