Problem 19 - Algebra 2

Question

Find the exact value of each expression. \(\sin \left(-15^{\circ}\right)\)

Step-by-Step Solution

Answer

\(\sin(-15^{\circ}) = \frac{-\sqrt{6} + \sqrt{2}}{4}\)

Step 1: Use the Sine Angle Addition Formula

The angle \(-15^{\circ}\) can be written as \\( -15^{\circ} = -45^{\circ} + 30^{\circ} \).Hence, we can use the sine angle addition formula:\[\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\]Let \(a = -45^{\circ}\) and \(b = 30^{\circ}\). We need to find \\(\sin(-15^{\circ}) = \sin(-45^{\circ} + 30^{\circ})\).

Step 2: Identify Known Trigonometric Values

We know:- \( \sin(30^{\circ}) = \frac{1}{2} \)- \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)- \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)- \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)These values will be used to calculate \( \sin(-45^{\circ}) = -\sin(45^{\circ}) \) and \( \cos(-45^{\circ}) = \cos(45^{\circ}) \).

Step 3: Substitute and Calculate

Substitute the known values into the sine addition formula:\[\sin(-15^{\circ}) = \sin(-45^{\circ}) \cos(30^{\circ}) + \cos(-45^{\circ}) \sin(30^{\circ})\]Substitute the values:\[\sin(-15^{\circ}) = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)\]Simplify:\[= -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\]

Step 4: Simplify the Expression

Combine the fractions:\[\sin(-15^{\circ}) = \frac{-\sqrt{6} + \sqrt{2}}{4}\]This is the exact value of \( \sin(-15^{\circ}) \).

Key Concepts

Sine Angle Addition Formula

The sine angle addition formula is a powerful tool in trigonometry. This formula helps in finding the sine of the sum or difference of two angles. It is expressed as \( \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \). It also applies to subtracting angles: \( \sin(a - b) = \sin(a) \cos(-b) + \cos(a) \sin(-b) \), using the properties of trigonometric functions that \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \). When describing these using simple words: sin of sum equals product of sin and cos, plus the product of cos and sin, when handled correctly with positive or negative signs as in original angles.

In the given exercise, the angle \(-15^{\circ}\) was broken into \(-45^{\circ} + 30^{\circ}\). This breakdown makes it possible to apply the sine angle addition formula. By substituting appropriate known values, you can find results that express the sine of an angle that might not initially come with easy intuitive exact values.

Exact Trigonometric Values

Trigonometric values for specific common angles are widely used in geometry and trigonometry. Angles such as \(30^{\circ}, 45^{\circ},\) and \(60^{\circ}\) have trigonometric ratios that are often memorized or easily accessible as they recur frequently. Below are some key values you should remember:

• \( \sin(30^{\circ}) = \frac{1}{2} \)
• \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)
• \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)
• \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)

For negative angles, note that \( \sin(-x) = -\sin(x) \) implies that sine is an odd function, and \( \cos(-x) = \cos(x) \) demonstrates cosine as an even function.

These fundamental truths allow the conversion from negative angles to their positive equivalents, simplifying the evaluation of trigonometric expressions.

Angle Identities

Trigonometric identities involve relationships that are true for angles across all ranges. These identities play a critical role in simplifying expressions and solving equations.

There are angle sum and difference identities, as seen in the angle addition formula. In these, the trigonometric functions maintain their relationships through sum or difference transformations. More generally:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Even-Odd Identities: Sine and tangent are odd functions, \( \sin(-x) = -\sin(x) \), \( \tan(-x) = -\tan(x) \), and cosine is even, \( \cos(-x) = \cos(x) \).

The use and understanding of these identities enable flexibility in handling a variety of trigonometric problems.

In tackling such exercises, knowing these identities aids in expressing complicated angles and achieving exact values, making trigonometry more approachable.