Problem 5 - Precalculus: Functions and Graphs

Question

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\sec \theta=\frac{5}{4} ; \quad 0^{\circ}<\theta<90^{\circ}$$

Step-by-Step Solution

Answer

\( \sin(\theta / 2) = \frac{\sqrt{10}}{10} \), \( \cos(\theta / 2) = \frac{3\sqrt{10}}{10} \), \( \tan(\theta / 2) = \frac{1}{3} \)."

Step 1: Identify the given information

We are given that \( \sec \theta = \frac{5}{4} \) and \( 0^{\circ} < \theta < 90^{\circ} \). The secant of an angle is the reciprocal of the cosine, hence \( \cos \theta = \frac{4}{5} \).

Step 2: Find \( \sin \theta \)

Use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).Substitute \( \cos \theta = \frac{4}{5} \) into the equation:\[ \sin^2 \theta + \left( \frac{4}{5} \right)^2 = 1 \]\[ \sin^2 \theta + \frac{16}{25} = 1 \]\[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \]\[ \sin \theta = \frac{3}{5} \] (since \( \sin \theta > 0 \) as \( 0^{\circ} < \theta < 90^{\circ} \)).

Step 3: Calculate \( \sin (\theta / 2) \)

Use the half-angle identity for sine: \( \sin (\theta / 2) = \sqrt{\frac{1 - \cos \theta}{2}} \).Substitute \( \cos \theta = \frac{4}{5} \) into the identity:\[ \sin (\theta / 2) = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10} \].

Step 4: Calculate \( \cos (\theta / 2) \)

Use the half-angle identity for cosine: \( \cos (\theta / 2) = \sqrt{\frac{1 + \cos \theta}{2}} \).Substitute \( \cos \theta = \frac{4}{5} \) into the identity:\[ \cos (\theta / 2) = \sqrt{\frac{1 + \frac{4}{5}}{2}} = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10} \].

Step 5: Calculate \( \tan (\theta / 2) \)

Use the half-angle identity: \( \tan (\theta / 2) = \frac{\sin (\theta)}{1 + \cos (\theta)} \).Substitute \( \sin \theta = \frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \):\[ \tan (\theta / 2) = \frac{\frac{3}{5}}{1 + \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{9}{5}} = \frac{3}{9} = \frac{1}{3} \].

Key Concepts

half-angle identities

Half-angle identities are a set of trigonometric identities that allow us to express trigonometric functions of half-angles in terms of the full angle. They are especially useful for solving problems involving angles that are not directly given, such as when you need to find the sine, cosine, or tangent of half of a known angle.

The identities are:

• \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \)
• \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \)
• \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} \)

These identities stem from the double angle formulas and help in determining the values of trigonometric functions where the angle is halved. This is extremely useful when you've determined \(\sin \theta\) and \(\cos \theta\) and need the half-angle values. In the exercise, they are applied to calculate \(\sin(\theta/2)\), \(\cos(\theta/2)\), and \(\tan(\theta/2)\) from \(\cos \theta\).

Pythagorean identity

The Pythagorean identity is one of the fundamental relations in trigonometry, playing a crucial role similar to its geometric counterpart, the Pythagorean theorem.

The identity is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This relationship shows how the squares of the sine and cosine of any angle add up to 1, a consequence directly derived from the properties of a right triangle or the unit circle. The unit circle definition of sine and cosine involves the hypotenuse having a constant length of 1.

In the given exercise, you utilize this identity to find \(\sin \theta\) given \(\cos \theta\). By substituting \(\cos \theta = \frac{4}{5}\) into the identity, we rearrange to solve \(\sin^2 \theta\) which results in \(\sin \theta = \frac{3}{5}\), fulfilling the identity since we know \(\sin \theta \) should be positive in the first quadrant.

trigonometric functions

Trigonometric functions are the building blocks of trigonometry, frequently used to relate angles with ratios of certain sides in right triangles or coordinates on the unit circle. The primary functions include sine, cosine, and tangent:

• **Sine (\( \sin \)):** Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine (\( \cos \)):** Represents the ratio of the adjacent side to the hypotenuse.
• **Tangent (\( \tan \)):** Represents the ratio of the opposite side to the adjacent side, or equivalently, \( \frac{\sin \theta}{\cos \theta} \).

These functions are periodic and can be explored further with identities and formulas to solve various angular relationships in trigonometric problems. In the problem, we start with \(\sec \theta\) which is the reciprocal of \(\cos \theta\), and are then able to find \(\sin \theta\) and move further to compute half-angle values.

secant and cosine relationship

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. This relationship is given by:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This identity is helpful when converting a problem stated in terms of secant back into a more familiar form using cosine.

For example, if \( \sec \theta = \frac{5}{4} \) as given in the exercise, we can immediately deduce:

• \( \cos \theta = \frac{4}{5} \)

Understanding this relationship allows one to determine the other trigonometric functions, such as obtaining \(\sin \theta\) via the Pythagorean identity. This initial conversion is critical in making subsequent calculations simpler and unlocking access to other trigonometric insights using established identities.