Problem 14 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \sin \left(60^{\circ}-45^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression \(\sin \left(60^{\circ}-45^{\circ}\right)\) is \((\sqrt{6} - \sqrt{2})/4\).

Step 1: Apply the Identity

Employing the trigonometric identity, the expression can be written as \(\sin \left(60^{\circ}-45^{\circ}\right) = \sin 60^{\circ} \cos 45^{\circ} - \cos 60^{\circ} \sin 45^{\circ}\).

Step 2: Substitute the Values

Since we know the standard values for sin and cos for \(45^{\circ}\) and \(60^{\circ}\) respectively, substitute them into the expression. \(\sin 60^{\circ} = \sqrt{3}/2\), \(\cos 60^{\circ} = 1/2\), \(\sin 45^{\circ} = \cos 45^{\circ} = \sqrt{2}/2\). The expression then becomes \((\sqrt{3}/2 * \sqrt{2}/2) - (1/2 * \sqrt{2}/2)\). This simplifies to \(\sqrt{6}/4 - \sqrt{2}/4\).

Step 3: Simplify the Expression

Since these two terms have the same denominator, they can be combined. This gives us the final simplified expression, \((\sqrt{6} - \sqrt{2})/4\).

Key Concepts

sum and difference identities

When working with trigonometric expressions, sum and difference identities are powerful tools. They allow us to express trigonometric functions of sums or differences of angles in terms of products of trigonometric functions of individual angles. For example, the identity for

• sum is \[\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\].
• difference is \[\sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b)\].

Understanding and using these identities simplifies the process of finding exact values of trigonometric expressions for angles that are sums or differences without needing a calculator. Applying these identities can help break down complex angles into familiar components, which makes solving trigonometric problems more approachable for students.
In our exercise, we use the difference identity for sine to rewrite \(\sin(60^\circ - 45^\circ)\)to more manageable terms involving well-known angles. This strategy highlights the versatility of trigonometric identities in simplifying expressions.

trigonometric functions

Trigonometric functions are mathematical functions related to angles of triangles and the unit circle. They primarily include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are foundational in trigonometry.
Each function has a specific meaning:

• \(\sin\) of an angle is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• \(\cos\) is the ratio of the adjacent side to the hypotenuse.
• \(\tan\) is the ratio of the opposite side to the adjacent side, or \(\tan = \sin/\cos\).

In the unit circle, an angle at the center of the circle yields coordinates \((x, y)\), where \(x = \cos\theta\) and \(y = \sin\theta\).
For our problem, understanding these functions is crucial because they help translate the angle difference into a form where you can use known values, like \(\sin 60^\circ\) and \(\cos 45^\circ\). Having the foundational knowledge of these functions gives us the tools to tackle many trigonometric problems efficiently.

exact values of trigonometric expressions

Trigonometry frequently involves determining exact values of trigonometric expressions, particularly for common angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\). These values are typically memorized due to their frequent use:

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

Having these values at your fingertips can expedite solving problems like \(\sin(60^\circ - 45^\circ)\) by allowing substitution into sum and difference identities.
In our example, substituting these known values transforms the expression into a rational form with whole numbers under square roots, simplifying to \(\frac{\sqrt{6} - \sqrt{2}}{4}\). This problem-solving method highlights the importance of remembering exact values for key angles, aiding in efficient and accurate calculations.