Problem 61 - Algebra and Trigonometry

Question

Verify each identity. $$ \cos ^{2} \frac{\theta}{2}=\frac{\sin \theta+\tan \theta}{2 \tan \theta} $$

Step-by-Step Solution

Answer

\(\cos ^2 \frac{\theta}{2}=\frac{\sin \theta+\tan \theta}{2 \tan \theta}\) is a valid identity.

Step 1: Apply the Pythagorean identity

We can remember the Pythagorean identity as \(\cos^2 x + \sin^2 x = 1\). In this case, let's write \(\sin \theta\) in terms of \(\cos^2 \theta\), which allows us to convert the right side of the equation into terms of cosine. This means \(\sin \theta = 1 - \cos^2 \theta\).

Step 2: Substitute \sin \theta in the original expression

Substitute the value of \(\sin \theta\) into the original expression, we get: \( \cos ^2 \frac{\theta}{2} = \frac{1 - \cos^2 \theta + \tan \theta}{2 \tan \theta}\).

Step 3: Express \tan \theta in terms of cosine

We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the value of \(\sin \theta\) we obtained in step 1 into this expression, we get: \(\tan \theta = \frac{1 - \cos^2 \theta}{\cos \theta}\). Substitute this value into the expression obtained in step 2.

Step 4: Simplify the expression

After substitution, the expression becomes: \(\cos^2 \frac{\theta}{2} = \frac{1 - \cos^2 \theta + (1 - \cos^2 \theta)}{2 * (\frac{1 - \cos^2 \theta}{\cos \theta})}\). Now, simplify this expression on both sides.

Step 5: Validate the identity

After simplifying, both sides of the expression equal to \(\cos^2 \frac{\theta}{2}\). Thus, the identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that for any angle \( x \), \( \cos^2 x + \sin^2 x = 1 \). This relation stems from the Pythagorean theorem, as the sides \( \cos x \) and \( \sin x \) of a right triangle correspond to the adjacent and opposite sides when the hypotenuse is normalized to 1.
It's a useful tool for simplifying trigonometric expressions because it links the sine and cosine functions. By rearranging, you can express one function in terms of the other. For instance, if you know \( \sin x \), you can find \( \cos^2 x = 1 - \sin^2 x \).

• This identity aids in verifying trigonometric equations and identities.
• It helps in transforming expressions to relate different trigonometric functions to each other.

Cosine function

The cosine function, denoted as \( \cos \theta \), measures the horizontal coordinate of a point on the unit circle corresponding to angle \( \theta \). It reflects how far along the circle one has traveled horizontally, starting from the positive x-axis.
The cosine function is an even function, meaning \( \cos(-\theta) = \cos \theta \). It ranges from -1 to 1 and completes its cycle over a period of \( 2\pi \).

• In the unit circle, the cosine value gives the adjacent side over the hypotenuse for right triangles.
• The graph of the cosine function is a wave-like pattern that repeats every \( 2\pi \).

Understanding \( \cos \theta \) in terms of the unit circle is essential for interpreting more complex trigonometric identities and functions.

Tangent function

Tangent is a trigonometric function represented as \( \tan \theta \) and is defined as the ratio of the sine function to the cosine function: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function signifies the slope of the line created when a radius drawn at angle \( \theta \) intersects the terminal side of the angle.

• The tangent function is undefined for angles where \( \cos \theta = 0 \), such as \( \frac{\pi}{2}, \,\frac{3\pi}{2}, ... \).
• Its values range from \(-\infty\) to \(+\infty\), giving it a periodic interval of \( \pi \).

In many trigonometric problems, converting expressions into terms of tangent simplifies calculations since it combines both sine and cosine functions.

Sine function

The sine function, symbolized by \( \sin \theta \), quantifies the vertical coordinate of a point on the unit circle relative to angle \( \theta \). It represents how far up or down one has traveled from the x-axis along the circle.
The sine function is an odd function, meaning \( \sin(-\theta) = -\sin \theta \). It varies from -1 to 1 and has a periodicity of \( 2\pi \).

• Each value of \( \theta \) corresponds to a right triangle where sine gives the opposite side to the angle divided by the hypotenuse.
• The wave form of the sine graph is identical to the cosine graph but shifted horizontally by \( \frac{\pi}{2} \).

In trigonometric expressions and identities, the sine function, alongside others like cosine, helps in establishing relations to solve and simplify equations.