Problem 6 - Algebra 2 and Trigonometry

Question

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=270^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

\(\sin(270^{\circ} - 60^{\circ}) = -\frac{1}{2}\) and \(\sin(270^{\circ} + 60^{\circ}) = -\frac{1}{2}\).

Step 1: Recall the Sine Angle Difference Identity

The identity for the sine of the difference of two angles is given by the formula: \( \sin(A-B) = \sin A \cos B - \cos A \sin B \) . We will use this identity to find \( \sin(270^{\circ} - 60^{\circ}) \).

Step 2: Substitute Angles into Identity

Substitute \(A = 270^{\circ}\) and \(B = 60^{\circ}\) into the identity: \( \sin(270^{\circ} - 60^{\circ}) = \sin 270^{\circ} \cos 60^{\circ} - \cos 270^{\circ} \sin 60^{\circ} \).

Step 3: Evaluate Trigonometric Functions

The trigonometric values needed are: \(\sin 270^{\circ} = -1\), \(\cos 270^{\circ} = 0\), \(\cos 60^{\circ} = \frac{1}{2}\), and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).

Step 4: Plug Values into Formula

Plug in the calculated values: \( \sin(270^{\circ} - 60^{\circ}) = (-1)(\frac{1}{2}) - (0)(\frac{\sqrt{3}}{2}) = -\frac{1}{2} - 0 = -\frac{1}{2} \).

Step 5: Recall the Sine Angle Sum Identity

The identity for the sine of the sum of two angles is given by the formula: \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). We will use this identity to find \( \sin(270^{\circ} + 60^{\circ}) \).

Step 6: Substitute Angles into Identity for Sum

Substitute \(A = 270^{\circ}\) and \(B = 60^{\circ}\) into the identity: \( \sin(270^{\circ} + 60^{\circ}) = \sin 270^{\circ} \cos 60^{\circ} + \cos 270^{\circ} \sin 60^{\circ} \).

Step 7: Evaluate Sine Angle Sum Formula

Using the same trigonometric values as before: \( \sin(270^{\circ} + 60^{\circ}) = (-1)(\frac{1}{2}) + (0)(\frac{\sqrt{3}}{2}) = -\frac{1}{2} + 0 = -\frac{1}{2} \).

Key Concepts

Sine Angle Difference

The sine angle difference identity is a handy formula used when you need to find the sine of the difference between two angles. This identity is mathematically expressed as:

• \( \sin(A-B) = \sin A \cos B - \cos A \sin B \)

In our example, the angles in question are \(A = 270^{\circ}\) and \(B = 60^{\circ}\). This formula lets us express the sine of \(270^{\circ} - 60^{\circ}\) as a combination of simpler trigonometric functions.
First, it's essential to substitute the angles into the formula, which becomes:

• \( \sin(270^{\circ} - 60^{\circ}) = \sin 270^{\circ} \cos 60^{\circ} - \cos 270^{\circ} \sin 60^{\circ} \)

After that, calculate each trigonometric value:

• \( \sin 270^{\circ} = -1 \)
• \( \cos 270^{\circ} = 0 \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

Once you substitute these values into the identity, the result is \( -\frac{1}{2} \). This process breaks a complex problem into smaller, manageable parts, showcasing the power of trigonometric identities.

Sine Angle Sum

The sine angle sum identity is the cousin of the sine angle difference identity. It helps calculate the sine of the sum of two angles with the formula:

• \( \sin(A+B) = \sin A \cos B + \cos A \sin B \)

Applying this to our given angles of \(A = 270^{\circ}\) and \(B = 60^{\circ}\), we derive:

• \( \sin(270^{\circ} + 60^{\circ}) = \sin 270^{\circ} \cos 60^{\circ} + \cos 270^{\circ} \sin 60^{\circ} \)

The trigonometric values previously calculated are still valid, and substituting them into the sine angle sum identity gives:

• \( \sin(270^{\circ} + 60^{\circ}) = (-1)\left(\frac{1}{2}\right) + (0)\left(\frac{\sqrt{3}}{2}\right) = -\frac{1}{2} + 0 \)

Hence, the final result for the sine angle sum is also \( -\frac{1}{2} \). This consistent result across different operations highlights how powerful understanding these identities can be in evaluating trigonometric expressions.

Evaluate Trigonometric Functions

Evaluating trigonometric functions can sometimes seem daunting, but breaking them down into recognizable and easy steps makes the task manageable.
Trigonometric functions like sine and cosine have standard values at specific angle points on the unit circle. These angles include multiples of \(90^{\circ}\) and \(60^{\circ}\) among others. Learning these values is crucial:

• \( \sin 270^{\circ} = -1 \)
• \( \cos 270^{\circ} = 0 \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

Using these known values, you can easily substitute into larger formulas such as those provided by the sine angle difference and angle sum identities.
By simplifying and solving these identities with known values, complex problems become understandable and solvable. Mastering these fundamentals allows for greater flexibility in handling a variety of trigonometric problems.