Problem 61 - 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. \( 112^\circ 30^\prime \)

Step-by-Step Solution

Answer

Use the provided step-by-step solution to calculate the sine, cosine, and tangent of the angle \( 112^\circ 30^\prime \) using the half-angle formulas.

Step 1: Convert to Decimal Degrees

First, convert the angle from degrees and minutes to decimal degrees. One minute is \(\frac{1}{60}\) of a degree, so\( 112^\circ 30^\prime = 112.5^\circ \). Thus, \( \theta = 112.5^\circ \).

Step 2: Find Cosine of the Angle

Next, find the cosine of \( \theta \), \( \cos \theta = \cos 112.5^\circ \), using a calculator or trigonometric table.

Step 3: Apply Half-Angle Formulas

Now we will apply the half-angle formulas to calculate the sine, cosine, and tangent of half the angle. Note the ambiguity in the sign. The sign will depend on the quadrant in which \( \frac{\theta}{2} \) is located.

Step 4: Determine the Correct Signs

Since \( 112.5 / 2 = 56.25 \) degrees is in the first quadrant, all signs will be positive. Calculate each function by plugging the value of \( \cos \theta \) into the equations.

Step 5: Simplify the Results

Finally, simplifying the results will give us the exact values for sine, cosine, and tangent of half the angle.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationship between the angles and sides of triangles, particularly right-angled triangles. It is essential for solving a wide range of problems across various fields, including physics and engineering.
Trigonometry relies on specific functions, namely sine, cosine, and tangent, that help calculate unknown elements of a triangle if certain angles or side lengths are known. Each trigonometric function provides a different ratio between sides of a right triangle, which can enable the calculation of angles or the other sides.
Understanding and applying trigonometry becomes invaluable in fields that involve measurement of space and angles, such as architecture, navigation, and even astronomy. Its application stretches from simple geometry problems to complex calculus and beyond. The use of equations like the half-angle formulas further ties into more advanced trigonometry, allowing the simplification and resolution of angles which are otherwise cumbersome to deal with in a raw numerical form.

Decimal Degrees

Decimal degrees provide a way of expressing angles using a decimal system rather than the traditional degrees, minutes, and seconds format. This method simplifies many calculations by converting angles into a single numerical value.
To convert from degrees and minutes to decimal degrees, it's important to remember that one minute is equal to \(\frac{1}{60}\) of a degree. Therefore, an angle, for example, 112 degrees and 30 minutes, would be converted to decimal degrees as follows:

• Take the full degrees: 112
• Convert the minutes into a fraction of a degree: 30 minutes or 30/60 = 0.5 degrees
• Add these together to get 112.5 degrees

Transforming angles into decimal degrees simplifies computations, especially when using calculators or computer software. The precision of decimal degrees helps in accurately applying trigonometric identities and other mathematical equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. They are tools that provide simplifications necessary for solving trigonometric equations and simplifying expressions.
Among these, the half-angle formulas are particularly helpful when finding the sine, cosine, and tangent of an angle that is half of another given angle. For example, if you have an angle \( \theta = 112.5^{\circ}\) and need to find the trigonometric values for \( \frac{\theta}{2} \), you need half-angle formulas.
Let’s break down the half-angle identities:

• For sine: \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)
• For cosine: \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
• For tangent: \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \) or \( \frac{\sin \theta}{1 + \cos \theta} \)

In applying these formulas, the sign depends on the quadrant where the angle lies. After determining the correct cosine, sine, or tangent function of the angle \( \theta \), simply plug those into the half-angle formulas to find the desired values.