Problem 21 - Algebra 2

Question

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) 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 3.

Step 1: Find the Value of Sin θ

We know that \(\sin^{2}\theta = 1-\cos^{2}\theta\). Given that \(\cos\theta = -\frac{4}{5}\), we can calculate \(\sin\theta\) by first finding \(\sin^{2}\theta = 1-(-\frac{4}{5})^2 = 1-\frac{16}{25} =\frac{9}{25}\). Since we know that \(\theta\) lies in the second quadrant where sine is positive, \(\sin\theta\) can be taken as \(\sqrt{\sin^{2}\theta}\), which equals \(\frac{3}{5}\).

Step 2: Calculate Cos θ/2 Using Half-Angle Formula

The half angle formula for cosine can be written as: \(\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}\). If we substitute the given \(\cos\theta=-\frac{4}{5}\) into this formula, it becomes: \(\cos\frac{\theta}{2} = \pm\sqrt{\frac{1-\frac{4}{5}}{2}} = \pm\sqrt{\frac{1}{10}}\).In the formula for half an angle, the sign before the square root depends on the quadrant of the angle. Since \(90^{\circ}<\theta<180^{\circ}\), then \(45^{\circ}<\frac{\theta}{2}<90^{\circ}\), which lies in the first quadrant where cosine is positive. Therefore, take the positive root, giving us: \( \cos\frac{\theta}{2} = \sqrt{\frac{1}{10}}\).

Step 3: Calculate Sin θ/2 Using Half-Angle Formula

The half angle formula for sine can be written as: \(\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}\). Similarly, substituting the given \(\cos\theta= -\frac{4}{5}\) into this formula, it becomes: \(\sin\frac{\theta}{2} = \pm\sqrt{\frac{1+\frac{4}{5}}{2}} = \pm\sqrt{\frac{9}{10}}\). Also, since \(\frac{\theta}{2}\) lies in the first quadrant where sine is positive, we choose the positive root, giving us: \(\sin\frac{\theta}{2} = \sqrt{\frac{9}{10}}\).

Step 4: Calculate Tan θ/2

Finally, we calculate \(\tan\frac{\theta}{2}\), which is equal to \(\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}}\). From the earlier steps, we have \(\cos\frac{\theta}{2} = \sqrt{\frac{1}{10}}\) and \(\sin\frac{\theta}{2} = \sqrt{\frac{9}{10}}\), hence \(\tan\frac{\theta}{2} = \frac{\sqrt{\frac{9}{10}}}{\sqrt{\frac{1}{10}}} = 3\).

Key Concepts

Half-Angle Identities

Understanding trigonometric identities is crucial in trigonometry. Half-angle identities are a special set that help find the trigonometric function values of half an angle. These identities are particularly useful when solving problems where one needs to determine values like \(\cos \frac{\theta}{2}\), \(\sin \frac{\theta}{2}\), or \(\tan \frac{\theta}{2}\). Formulae used include:

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

In these formulas, the plus or minus sign is determined by the quadrant in which the resulting angle lies. This makes half-angle identities particularly handy when working with second quadrant angles, as they allow us to simplify calculations by considering only positive or negative signs based on our quadrant analysis.

Trigonometric Functions

When dealing with angles, it’s important to know the main trigonometric functions: sine, cosine, and tangent. They represent relationships between the angles and sides of a right triangle. Each function provides vital information:

• **Sine (\(\sin\))** measures the ratio of the opposite side to the hypotenuse.
• **Cosine (\(\cos\))** measures the ratio of the adjacent side to the hypotenuse.
• **Tangent (\(\tan\))** is the ratio of sine to cosine or of the opposite side to the adjacent side.

Using these functions, you gain insights into an angle’s properties. For example, given \(\cos \theta = -\frac{4}{5}\), we find \(\sin \theta\) using Pythagorean identity \(\sin^2\theta = 1 - \cos^2\theta\). In trigonometry, using identities effectively allows you to transition between functions and simplify expressions.

Second Quadrant Angle Analysis

Determining the properties of angles in the second quadrant is essential. Angles in this area range from \(90^{\circ}\) to \(180^{\circ}\). They have unique characteristics:

• **Cosine is negative:** Since angles fall within quadrant II, \(\cos \theta\) takes a negative value.
• **Sine is positive:** In contrast, \(\sin \theta\) remains positive, providing a distinct identity for second quadrant angles.
• **Tangent is negative:** With positive sine and negative cosine, \(\tan \theta\) is negative in this quadrant.

These properties influence how you apply trigonometric identities. In the given exercise, knowing that \(\theta\) is in the second quadrant guides the selection of positive sqrt for sine and which angle to consider for further identities. It’s important to check these attributes whenever dealing with trigonometric expressions in the second quadrant.