Problem 44 - Algebra and Trigonometry

Question

Inverse Trigonometric Functions on a Calculator Most calculators do not have keys for sec \(^{-1}, \mathrm{csc}^{-1},\) or cot \(^{-1}\) . Prove the following identities, and then use these identities and a calculator to find \(\sec ^{-1} 2, \mathrm{csc}^{-1} 3,\) and \(\cot ^{-1} 4 .\) $$\begin{array}{ll}{\sec ^{-1} x=\cos ^{-1}\left(\frac{1}{x}\right),} & {x \geq 1} \\ {\csc ^{-1} x=\sin ^{-1}\left(\frac{1}{x}\right),} & {x \geq 1} \\\ {\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{x}\right),} & {x>0}\end{array}$$

Step-by-Step Solution

Answer

\(\sec^{-1} 2 = 60°\), \(\csc^{-1} 3 \approx 19.47°\), \(\cot^{-1} 4 \approx 14.04°\).

Step 1: Understanding the Task

We need to find the angles corresponding to the inverse secant, cosecant, and cotangent functions using a calculator that provides keys for inverse sine, cosine, and tangent functions only.

Step 2: Substitute and Validate Identities

Since calculators don't have keys for \(\sec^{-1}, \csc^{-1},\) or \(\cot^{-1}\), we use the given identities: \(\sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right)\), \(\csc^{-1} x = \sin^{-1}\left(\frac{1}{x}\right)\), and \(\cot^{-1} x = \tan^{-1}\left(\frac{1}{x}\right)\). They show that the inverse functions can be rewritten using the simpler inverse trigonometric functions available on calculators.

Step 3: Calculate \(\sec^{-1} 2\)

According to the identity, \(\sec^{-1} 2 = \cos^{-1}\left(\frac{1}{2}\right)\). Use a calculator to find \(\cos^{-1}(0.5)\). This will give an angle in radians or degrees. In degrees, \(\cos^{-1}(0.5)\) is 60°.

Step 4: Calculate \(\csc^{-1} 3\)

Using the identity \(\csc^{-1} 3 = \sin^{-1}\left(\frac{1}{3}\right)\), use the calculator to find \(\sin^{-1}(0.333...)\). The angle can be found in degrees or radians. Approximate this to be around 19.47° when calculated in degrees.

Step 5: Calculate \(\cot^{-1} 4\)

The identity \(\cot^{-1} 4 = \tan^{-1}\left(\frac{1}{4}\right)\) helps. Use the calculator to find \(\tan^{-1}(0.25)\), which gives the angle. In degrees, \(\tan^{-1}(0.25)\) is approximately 14.04°.

Key Concepts

Inverse Secant Function

The inverse secant function, denoted as \( \sec^{-1}(x) \), is used to find the angle whose secant is \( x \). Secant is the reciprocal of the cosine function, defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
This means the inverse secant can be expressed in terms of the inverse cosine function because most calculators do not have a direct \( \sec^{-1} \) function button.When you're given an \( x \), such as \( \sec^{-1}(2) \), you rewrite it using the identity:

• \( \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \)

This identity utilizes the relationship between secant and cosine. For example, \( \sec^{-1}(2) \) becomes \( \cos^{-1}\left(\frac{1}{2}\right) \). You can quickly calculate this using your calculator's \( \cos^{-1} \) function, resulting in an angle of 60° when considered in degrees.

Inverse Cosecant Function

The inverse cosecant function, \( \csc^{-1}(x) \), determines the angle whose cosecant is \( x \). Similar to secant, cosecant is the reciprocal of sine, described by \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
To simplify calculations, we express \( \csc^{-1} \) using \( \sin^{-1} \) because it is commonly available on calculators.The identity used is:

• \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \)

Applying this identity to \( \csc^{-1}(3) \), we convert it to \( \sin^{-1}\left(\frac{1}{3}\right) \). This translates into using the calculator to find \( \sin^{-1}(0.333...) \), which roughly equals 19.47° in degrees.

Inverse Cotangent Function

The inverse cotangent function, or \( \cot^{-1}(x) \), helps find the angle whose cotangent equals \( x \). Cotangent is the reciprocal of the tangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
Since many calculators lack a \( \cot^{-1} \) key, we rely on expressing it through \( \tan^{-1} \), a more accessible function.The corresponding identity is:

• \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \)

Using this for \( \cot^{-1}(4) \), we find it equals \( \tan^{-1}\left(\frac{1}{4}\right) \). Calculating \( \tan^{-1}(0.25) \) via a calculator gives an angle of about 14.04° in degrees. This approach simplifies obtaining the inverse angles without needing the specific \( \cot^{-1} \) function.