Problem 59 - Precalculus with Limits

Question

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( 75^\circ \)

Step-by-Step Solution

Answer

The exact values of the sine, cosine, and tangent of \( 75^\circ \) are \( \sin 37.5^\circ = \sqrt{\frac{1 - \sqrt{3}}{2\sqrt{2}}}\), \( \cos 37.5^\circ = \sqrt{\frac{1 + \sqrt{3}}{2\sqrt{2}}}\), and \( \tan 37.5^\circ = \sqrt{2} - 1\), respectively.

Step 1: Identifying Half-Angle and Applying Formula

The first step is to identify the half-angle, which in this case is \( \frac{75}{2} = 37.5^\circ \). Then, apply half-angle formulas. The half-angle formula for sine is: \( \sin \frac{\theta}{2} = ± \sqrt{\frac{1 - \cos \theta}{2}} \). For cosine is: \( \cos \frac{\theta}{2} = ± \sqrt{\frac{1 + \cos \theta}{2}} \). The formula for tangent can be derived from sine and cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta}\).

Step 2: Calculating Cosine of Angle

Next, we need to calculate the cosine of \( 37.5^\circ \) which is \( \cos 37.5^\circ = \sqrt{\frac{1 + \sqrt{3}}{2\sqrt{2}}}\).

Step 3: Calculating Sine of Angle

Then we find the sine of \( 37.5^\circ \) which is \( \sin 37.5^\circ = \sqrt{\frac{1 - \sqrt{3}}{2\sqrt{2}}}\).

Step 4: Calculating Tangent of Angle

Finally, calculate the tangent of \( 37.5^\circ \). Because \( \tan \theta = \frac{\sin \theta}{\cos \theta}\), we have \( \tan 37.5^\circ = \frac{\sqrt{\frac{1 - \sqrt{3}}{2\sqrt{2}}}}{\sqrt{\frac{1 + \sqrt{3}}{2\sqrt{2}}}} = \sqrt{2} - 1\).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental to understanding angles and circles. These functions help us relate angles to ratios of sides in a right triangle.

In the realm of these functions:

• The sine function, noted as \( \sin \theta \), relates to the ratio of the opposite side over the hypotenuse.
• The cosine function, noted as \( \cos \theta \), involves the ratio of the adjacent side over the hypotenuse.
• The tangent function, noted as \( \tan \theta \), is the ratio of the sine to the cosine, or opposite over adjacent.

For example, when we aim to find the trigonometric values for a particular angle like \( 75^\circ \), we often rely on half-angle formulas. These formulas are especially useful for angles that are not commonly found in the unit circle or simple triangles.

Understanding these functions and how they are applied in calculations, like determining the sine, cosine, and tangent, can greatly enhance your ability to solve trigonometry problems.

The Process of Angle Calculation

Calculating angles often involves using known values and mathematical formulas to determine unknown ratios or angles. Specifically, using the half-angle formulas provides a precise way to calculate trigonometric functions of an angle formed by halving another angle.

In our case:

• The half-angle for \( 75^\circ \) is determined by taking \( \frac{75}{2} \), resulting in \( 37.5^\circ \).
• We then implement the half-angle formulas to find the exact trigonometric functions of \( 37.5^\circ \).

To calculate these values accurately, knowing the formulas is crucial:

• The formulation for sine at half the angle \( \theta \) is \(\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}\).
• The cosine formulation is \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\).
• The tangent can be derived using these or simplified to \( \tan \theta = \frac{\sin \theta}{\cos \theta}\).

Through these computations, we achieved an exact value for the sine and cosine of \( 37.5^\circ \), leading to our final solution for \( 75^\circ \).

Successfully calculating these angles requires understanding both the arithmetic and trigonometric nature of half-angle calculations.

Deriving Exact Values From Half-Angle Formulas

The beauty of using half-angle formulas lies in their ability to provide accurate exact values for angles that aren't directly accessible. For many angles, especially those not found on the unit circle, these formulas are indispensable.

Let's look at the sine and cosine for \( 37.5^\circ \) more closely:

• The calculated cosine \( \cos 37.5^\circ = \sqrt{\frac{1 + \sqrt{3}}{2\sqrt{2}}} \) is derived by substituting known values in the cosine half-angle formula.
• Similarly, the sine value \( \sin 37.5^\circ = \sqrt{\frac{1 - \sqrt{3}}{2\sqrt{2}}} \) emerges from the sine half-angle formula.

When it comes to calculating the tangent, combining these results gives us \( \tan 37.5^\circ = \sqrt{2} - 1 \), showing how interconnected these functions are.

Achieving exact values isn't just about substituting numbers into a formula. It involves a deep understanding of trigonometric identities and rationalizing expressions. These exact values are crucial for applications in solving equations and physics problems where precision is vital.