Problem 40 - Algebra 2

Question

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \tan \frac{\theta}{2} $$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{\theta}{2} \) is \( -\frac{1}{2} \).

Step 1: Calculate the sine of the angle

Using the Pythagorean identity, we can compute for the sine of the angle \( \theta \) by evaluating \( \sin \theta = - \sqrt{1-\cos^{2} \theta} = - \sqrt{1 - \left(\frac{3}{5}\right)^{2}} = -\frac{4}{5} \).

Step 2: Use the half-angle formula

The half-angle formula for the tangent of an angle is \( \tan(\frac{\theta}{2}) = \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} \). Given that \( \cos \theta = \frac{3}{5} \), we can substitute into this formula to compute for \( \tan(\frac{\theta}{2}) \).

Step 3: Evaluate the expression

Substituting \( \cos \theta = \frac{3}{5} \) into the half-angle formula gives \( \tan(\frac{\theta}{2}) = \sqrt{\frac{1-\frac{3}{5}}{1+\frac{3}{5}}} = \sqrt{\frac{2}{8}} = \frac{1}{2} \). To maintain the appropriate sign for the quadrant, the result is negative since the fourth quadrant has negative tangents, resulting in \( \tan(\frac{\theta}{2}) = -\frac{1}{2} \).

Key Concepts

Half-Angle Formulas

Half-angle formulas are essential in trigonometry for breaking down angles into their halves. They provide a way to compute trigonometric functions of half-angles in terms of the full angle. One common half-angle formula used is for the tangent function:
\[\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}\]
The plus or minus sign depends on the quadrant the angle falls into after being halved.

• For angles in the fourth quadrant, like the one in our problem, the tangent is negative.
• For angles between 0° and 180°, or first and third quadrants, tangent is positive in the first and negative in the third.

This is particularly useful not only for finding tangents but also because knowing one side of a right triangle can help infer others using these formulas.

Pythagorean Identities

Pythagorean identities are a cornerstone of trigonometry. They relate the squares of sine, cosine, and tangent to the number 1 and help us find one trigonometric function in terms of another. The primary Pythagorean identity is:
\[\sin^2 \theta + \cos^2 \theta = 1\]
From this identity, if we know the cosine, we can find the sine:

• By rearranging, we have \( \sin \theta = \pm \sqrt{1-\cos^2 \theta} \).
• The sign before the square root depends on the specific quadrant of the angle.

In the provided scenario, knowing \( \cos \theta = \frac{3}{5} \) helps us determine \( \sin \theta \). Since our angle \( \theta \) is in the fourth quadrant, where sine is negative, we evaluate it as \( \sin \theta = -\frac{4}{5} \). This step is critical for determining the behavior of the sine function depending on the angle's quadrant.

Trigonometric Quadrants

Understanding trigonometric quadrants is crucial when working with angles, especially when dealing with signs of trig functions. Trigonometric functions change their signs depending on which quadrant the angle is in:

• First Quadrant (0° to 90°): all trigonometric functions are positive.
• Second Quadrant (90° to 180°): only sine is positive.
• Third Quadrant (180° to 270°): only tangent is positive.
• Fourth Quadrant (270° to 360°): only cosine is positive.

In this problem, \( \theta \) was in the fourth quadrant, specifically between 270° to 360°, which means \( \cos \theta \) is positive, and \( \tan \theta \) and \( \sin \theta \) are negative.
This understanding ensures that when computing trigonometric functions using identities or formulas, such as half-angle formulas, the sign of the result aligns with the angle's quadrant, ensuring accuracy in any trigonometric analysis.