Problem 17 - Precalculus with Limits

Question

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( 105^\circ = 60^\circ + 45^\circ \)

Step-by-Step Solution

Answer

Therefore, the exact values are \(\sin 105^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}\), \(\cos 105^\circ = \frac{\sqrt{2} - \sqrt{6}}{4}\), and \(\tan 105^\circ = -\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}.\)

Step 1: Determine the Sine

Use the sine addition theorem, which is formulated as \( \sin (a+b) = \sin a \cdot \cos b + \cos a \cdot \sin b \). This theorem allows us to determine the sine of \( 105^\circ \) by using the sine and cosine values for \( 60^\circ \) and \( 45^\circ \). Thus, we have: \( \sin 105^\circ = \sin 60^\circ \cdot \cos 45^\circ + \cos 60^\circ \cdot \sin 45^\circ = \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} + \frac{1}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Step 2: Determine the Cosine

Next, we use the cosine addition theorem, \(\cos (a+b) = \cos a \cdot \cos b - \sin a \cdot \sin b \), to determine the cosine of \( 105^\circ \). Substituting the given angles yields: \(\cos 105^\circ = \cos 60^\circ \cdot \cos 45^\circ - \sin 60^\circ \cdot \sin 45^\circ = \frac{1}{2}\cdot\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{2} - \sqrt{6}}{4} \)

Step 3: Determine the Tangent

Lastly, we calculate the tangent of \(105^\circ \) using the definition \(\tan a = \frac{\sin a}{\cos a} \). Using the previously calculated sine and cosine values, we find: \( \tan 105^\circ = \frac{\sin 105^\circ}{\cos 105^\circ} = \frac{\frac{\sqrt{6} + \sqrt{2}}{4}}{\frac{\sqrt{2} - \sqrt{6}}{4}} = -\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\)

Key Concepts

Sine Addition Formula

To understand and apply the sine addition formula is very straightforward. This formula allows you to find the sine of the sum of two angles, say \( a \) and \( b \). It's expressed as \( \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \).

The sine addition formula essentially breaks down the problem into manageable parts using known values of sine and cosine. For the example of calculating \( \sin 105^\circ \), you can think of \( 105^\circ \) as the sum of \( 60^\circ \) and \( 45^\circ \). This approach is super helpful because sine and cosine values for these angles are quite common and can often be found directly or are memorized:

• \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \cos 60^\circ = \frac{1}{2} \)
• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)

By inserting these values into the formula, you arrive at:

\[ \sin 105^\circ = \sin 60^\circ \cdot \cos 45^\circ + \cos 60^\circ \cdot \sin 45^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \]

This process is intuitive once you practice with a variety of angles, so don't hesitate to work through some additional examples to build your confidence.

Cosine Addition Formula

The cosine addition formula is another essential trigonometric identity to master. It reads as follows: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \).

This formula helps you find the cosine of the sum of two angles by utilizing the individual cosine and sine values of those angles. In the example where you calculate \( \cos 105^\circ \), you treat \( 105^\circ \) as \( 60^\circ + 45^\circ \). This gives you a chance to utilize those familiar trigonometric values again. You would plug in:

• \( \cos 60^\circ = \frac{1}{2} \)
• \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)

By substituting in these values, you compute:

\[ \cos 105^\circ = \cos 60^\circ \cdot \cos 45^\circ - \sin 60^\circ \cdot \sin 45^\circ = \frac{\sqrt{2} - \sqrt{6}}{4} \]

Grasping this formula is about seeing the relationships between the separate angle components. With practice, understanding these relationships will become second nature.

Tangent Calculation

To calculate the tangent of an angle, it’s useful to remember that tangent is the ratio of sine to cosine. Mathematically, you can represent this as \( \tan a = \frac{\sin a}{\cos a} \).

When you already have the sine and cosine for a given angle, finding the tangent is a matter of simple division. Consider \( 105^\circ \) in our example where you have calculated both \( \sin 105^\circ \) and \( \cos 105^\circ \). With:

• \( \sin 105^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \)
• \( \cos 105^\circ = \frac{\sqrt{2} - \sqrt{6}}{4} \)

You perform a division to get the tangent:

\[ \tan 105^\circ = \frac{\frac{\sqrt{6} + \sqrt{2}}{4}}{\frac{\sqrt{2} - \sqrt{6}}{4}} = -\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} \]

When doing this, make sure to handle the negative signs and simplify the fraction as necessary. This process might involve rationalizing the denominator, which can be an interesting way to further refine your solution.