Problem 20 - Precalculus with Calculus Previews

Question

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin 345^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 345^{\circ} = \frac{\sqrt{2} - \sqrt{6}}{4} \)

Step 1: Identify the Formula

We can use the sum and difference identities for sine to evaluate \( \sin 345^{\circ} \). Noticing that \( 345^{\circ} = 360^{\circ} - 15^{\circ} \), we can use the identity for the sine of a difference: \[ \sin(360^{\circ} - \theta) = \sin 360^{\circ} \cos \theta - \cos 360^{\circ} \sin \theta \] which simplifies to: \( \sin(-\theta) = -\sin \theta \) due to the periodicity and co-function properties of standard angles.

Step 2: Evaluate the Components

For this identity, apply \( \theta = 15^{\circ} \):- \( \sin 360^{\circ} = 0 \)- \( \cos 360^{\circ} = 1 \)Thus:\[ \sin(345^{\circ}) = 0 \cdot \cos 15^{\circ} - 1 \cdot \sin 15^{\circ} \] This simplifies to \(-\sin 15^{\circ}\).

Step 3: Evaluate \( \sin 15^{\circ} \)

To calculate \( \sin 15^{\circ} \), use the angle subtraction identity \( \sin(45^{\circ} - 30^{\circ}) \):\[ \sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ} \] Using known values:- \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)- \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)- \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)- \( \sin 30^{\circ} = \frac{1}{2} \)Substitute these into the equation:\[ \sin 15^{\circ} = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2} \]

Step 4: Simplify the Expression

Simplify the expression for \( \sin 15^{\circ} \):\[ \sin 15^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \] Thus, \( \sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 5: Determine the Result

Since \( \sin 345^{\circ} = -\sin 15^{\circ} \), we substitute the expression found for \( \sin 15^{\circ} \):\[ \sin 345^{\circ} = -\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) \]Simplify this to:\[ \sin 345^{\circ} = \frac{-\sqrt{6} + \sqrt{2}}{4} \] or written as:\[ \sin 345^{\circ} = \frac{\sqrt{2} - \sqrt{6}}{4} \]

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, commonly denoted as \( \sin \). It relates to the ratio of the length of the opposite side to the hypotenuse in a right triangle. The sine function is essential for modeling periodic phenomena and understanding wave patterns.
It has a periodic nature with a period of \( 360^{\circ} \) or \( 2\pi \) radians. At \( 0^{\circ} \) and \( 360^{\circ} \), \( \sin \) takes the value of 0 due to its symmetry about these angles. This periodic nature also reflects that \( \sin(360^{\circ} + \theta) = \sin \theta \), and \( \sin(180^{\circ} - \theta) = \sin \theta \).
Additionally, it's important to notice that the sine function is an odd function, which means that \( \sin(-\theta) = -\sin \theta \). This property leads into various identities and transformations that simplify calculations in trigonometry.

Angle Sum and Difference Identities

The angle sum and difference identities are powerful tools in trigonometry that help us compute the sine (and other trigonometric functions) of complex angles using simpler known values.
These identities break down as follows:

• For sums: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• For differences: \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

These identities allow us to express tricky angles in terms of angles whose trigonometric values we may already know. For the sine of an angle like \( 345^{\circ} \), these identities typically help simplify the calculation to a difference, as in this case: \( 345^{\circ} = 360^{\circ} - 15^{\circ} \).
Thus, by knowing the sine values of standard angles such as \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \), we can break down more complex angles into these simpler parts.

Exact Trigonometric Values

Exact trigonometric values are key to solving problems without resorting to approximate computations, ensuring ultra-precise results.
For key angles like \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \), the values of sine, cosine, and tangent are often memorized:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)

Using these critical values in conjunction with the angle sum and difference identities, we can find precise values for non-standard angles. For example, calculating \( \sin 15^{\circ} \) via \( \sin(45^{\circ} - 30^{\circ}) \), results in a calculated exact value: \( \sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4} \). These exact calculations keep our answers precise and reliable in trigonometric problem-solving.