Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Use fundamental identities to write the first expression in terms of the second, for any acute angle \(\theta\). $$\tan \theta, \cos \theta$$

Step-by-Step Solution

Answer

\( \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \)

Step 1: Define tan in terms of sine and cosine

The tangent of an angle \( \theta \) is defined as the ratio of the sine and cosine of the angle. Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can proceed to express everything in terms of \( \cos \theta \).

Step 2: Express sin in terms of cos

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express sine in terms of cosine: \( \sin \theta = \sqrt{1 - \cos^2 \theta} \), assuming \( \theta \) is an acute angle and sine is positive.

Step 3: Write tan in terms of cos

Substitute the expression from Step 2 into the definition of tangent from Step 1: \[ \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \]. This expresses \( \tan \theta \) fully in terms of \( \cos \theta \).

Key Concepts

Tangent Function

The tangent function, often referred to as "tan," is a fundamental trigonometric function. It plays a crucial role in mathematics, physics, and engineering. The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the opposite side to the adjacent side. This can be expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

In terms of trigonometric identities, tangent can also be represented as the ratio of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This identity is very useful, particularly when rewriting trigonometric expressions in different forms.
For example, if you know \( \cos \theta \) and need to find \( \tan \theta \), you can substitute \( \sin \theta \) in terms of \( \cos \theta \) using the Pythagorean identity. This turns out to be particularly handy when solving equations or simplifying expressions that involve trigonometric functions.

Cosine Function

The cosine function, denoted as \( \cos \), is another essential trigonometric function. In the context of a right triangle, the cosine of an angle \( \theta \) is the ratio of the length of the adjacent side to the hypotenuse:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Cosine is fundamental in waves, oscillations, and circular motion studies. It's a periodic function and important in evaluating angles and constructing identities. It ranges from -1 to 1 and repeats every \( 360^\circ \) or \( 2\pi \) radians.
When exploring different identities, you often express other trigonometric functions in terms of cosine, as it's straightforward to manipulate. For instance, to find \( \tan \theta \) fully expressed in terms of \( \cos \theta \), you can use the identities given earlier in combination with sine.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone relationships in trigonometry. It states that for any angle \( \theta \), the squares of sine and cosine add up to 1:

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

This identity is derived from the Pythagorean theorem applied to a unit circle. This relationship allows us to express one trigonometric function in terms of another.
For instance, if you have the cosine of an angle and need to find the sine, you can rearrange this identity to find: \( \sin \theta = \sqrt{1 - \cos^2 \theta} \). This rearrangement is particularly useful when calculating the tangent function solely in terms of cosine. Thus, knowing any of these two ratios lets you find the third one using this identity.

Sine Function

The sine function is another primary trigonometric function, symbolized by \( \sin \). For an angle \( \theta \) in a right triangle, sine is defined as the ratio of the length of the opposite side to the hypotenuse:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

Like cosine, sine also ranges from -1 to 1 and is periodic, repeating every \( 360^\circ \) or \( 2\pi \) radians.
By using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), you can often interchange between sine and cosine. For acute angles (angles less than \( 90^\circ \)), the sine value is positive.
In practical problems, converting sine into expressions involving cosine using the Pythagorean identity helps simplify the evaluation of tangent or verify trigonometric expressions. Understanding sine deeply aids in solving real-world problems related to waves, sound, and light. Understanding how sine interacts with other trigonometric functions is vital for mastering trigonometric identities and their numerous applications.