Problem 85 - Precalculus

Question

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point. \(\sin \left(195^{\circ}\right)\)

Step-by-Step Solution

Answer

The exact value of \(\sin(195^{\circ})\) is \(-\frac{\sqrt{6} - \sqrt{2}}{4}\). This approximates to \(-0.2588\).

Step 1: Identify the Angle

The angle given is \(195^{\circ}\). Notice that \(195^{\circ}\) is in the third quadrant of the unit circle, where the sine function is negative.

Step 2: Express the Angle as a Sum or Difference

Rewrite \(195^{\circ}\) in terms of known angles. We can express it as \(180^{\circ} + 15^{\circ}\). This helps because we know the exact values of trigonometric functions for \(180^{\circ}\) and other key angles.

Step 3: Use the Sine Addition Formula

Use the sine addition formula: \[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \] For \(a = 180^{\circ}\) and \(b = 15^{\circ}\), calculate:\[ \sin(195^{\circ}) = \sin(180^{\circ} + 15^{\circ}) = \sin(180^{\circ})\cos(15^{\circ}) + \cos(180^{\circ})\sin(15^{\circ}) \] Substitute the known values: \(\sin(180^{\circ}) = 0\), \(\cos(180^{\circ}) = -1\).

Step 4: Calculate Values

Substitute the values into our formula:\[ \sin(195^{\circ}) = 0 \cdot \cos(15^{\circ}) + (-1) \cdot \sin(15^{\circ}) \]As a result, \[ \sin(195^{\circ}) = -\sin(15^{\circ}) \]

Step 5: Use Known Values for Sine of 15 Degrees

\(15^{\circ}\) can be expressed as \(45^{\circ} - 30^{\circ}\). Use the sine difference formula:\[ \sin(15^{\circ}) = \sin(45^{\circ} - 30^{\circ}) = \sin(45^{\circ})\cos(30^{\circ}) - \cos(45^{\circ})\sin(30^{\circ}) \]Using known values, find:\[ \sin(45^{\circ}) = \frac{\sqrt{2}}{2}, \cos(30^{\circ}) = \frac{\sqrt{3}}{2}, \cos(45^{\circ}) = \frac{\sqrt{2}}{2}, \sin(30^{\circ}) = \frac{1}{2} \]

Step 6: Calculate Sine of 15 Degrees

Substitute the values into the formula for \(\sin(15^{\circ})\):\[ \sin(15^{\circ}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \]Simplify:\[ \sin(15^{\circ}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \]

Step 7: Calculate the Final Result

Now that we know \(\sin(15^{\circ})\), we substitute it back:\[ \sin(195^{\circ}) = -\frac{\sqrt{6} - \sqrt{2}}{4} \]

Step 8: Confirm with a Calculator

Use a calculator to check \(\sin(195^{\circ})\). The calculator should confirm the result as approximately \(-0.2588\), matching the exact calculation: \[-\frac{\sqrt{6} - \sqrt{2}}{4} \approx -0.2588 \] to four decimal places.

Key Concepts

Sine Addition Formula

The sine addition formula is a mathematical expression used to find the sine of an angle that is expressed as the sum of two other angles. Let's break it down:

The formula you need to remember is:

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)

This means the sine of a combined angle \( a + b \) can be calculated using the sines and cosines of the individual angles \( a \) and \( b \).

It's like having a recipe for cooking. You mix together basic ingredients to make a new dish. In trigonometry, these ingredients are the values of sine and cosine for simpler angles that you already know. Consider angles like \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, \) and \( 90^\circ \), for which you can easily memorize the sine and cosine values.

Let's go over an example. If you need to find \( \sin(195^\circ) \), you can express it as the sum \( 180^\circ + 15^\circ \). By using the sine addition formula, you calculate:

• \( \sin(195^\circ) = \sin(180^\circ + 15^\circ) = \sin(180^\circ)\cos(15^\circ) + \cos(180^\circ)\sin(15^\circ) \)

By substituting known values for \( \sin(180^\circ) = 0 \) and \( \cos(180^\circ) = -1 \), you simplify the expression as

• \( 0 \cdot \cos(15^\circ) - 1 \cdot \sin(15^\circ) = -\sin(15^\circ) \)

This way, the sine addition formula helps unlock tricky calculations.

Unit Circle

The unit circle is an essential tool in trigonometry that helps visualize and understand the behavior of the sine and cosine functions. Imagine a circle with a radius of 1 unit centered at the origin of a coordinate plane. Each point on the circle can represent an angle in radians or degrees.

Here’s why it's useful:

• Angles are measured from the positive x-axis, moving counter-clockwise.
• As you rotate along the circle, the x-coordinate of a point gives you \( \cos(\theta) \) and the y-coordinate gives you \( \sin(\theta) \).

Think of the unit circle as your trigonometric dictionary. It provides quick access to the values of sine and cosine for many standard angles.

In our example with \( 195^\circ \), this angle is situated in the third quadrant. In that specific region, the sine values are negative because the y-coordinates of the unit circle are below the x-axis. These coordinates give us insight into the sign of our trigonometric functions:

• In Quadrant III, both sine and cosine are negative

This insight assists in predicting the behavior of sine and cosine without explicit calculation, merely through the visual aid of the circle.

Angle in Third Quadrant

Understanding an angle's position, specifically when it lies in the third quadrant, is crucial for determining the sign (positive or negative) of trigonometric functions.

The unit circle is divided into four quadrants:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative and cosine is positive.

When you have an angle like \( 195^\circ \) that lands in Quadrant III, it implies two things:

• The sine of this angle is negative.
• The cosine of this angle is also negative, which influences the signs when using addition or difference formulas.

This understanding is like having a roadmap. By knowing in which quadrant an angle sits, you can pinpoint whether its trigonometric ratios are positive or negative.

It's particularly helpful when you break down complex angles into sums or differences of known values (like \( 180^\circ + 15^\circ \) in this case), ensuring your calculated results hold the correct sign.