Problem 57 - Algebra 2

Question

Find the exact value of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. \(\cos \theta=\frac{5}{6} ; 0^{\circ}<\theta<90^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{5\sqrt{11}}{18}\), \(\cos 2\theta = \frac{7}{18}\), \(\sin \frac{\theta}{2} = \frac{\sqrt{3}}{6}\), \(\cos \frac{\theta}{2} = \frac{\sqrt{33}}{6}\).

Step 1: Use Pythagorean Identity

We have \( \cos \theta = \frac{5}{6} \). To find \( \sin \theta \), we use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta \): \[ \sin^2 \theta = 1 - \left(\frac{5}{6}\right)^2 = 1 - \frac{25}{36} = \frac{11}{36}. \] Thus, \( \sin \theta = \sqrt{\frac{11}{36}} = \frac{\sqrt{11}}{6} \).

Step 2: Find \( \sin 2\theta \)

Use the double angle formula for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substitute the known values: \[ \sin 2\theta = 2 \times \frac{\sqrt{11}}{6} \times \frac{5}{6} = \frac{10\sqrt{11}}{36} = \frac{5\sqrt{11}}{18}. \]

Step 3: Find \( \cos 2\theta \)

Use the double angle formula for cosine: \( \cos 2\theta = 2 \cos^2 \theta - 1 \). Substitute the known value: \[ \cos 2\theta = 2 \times \left(\frac{5}{6}\right)^2 - 1 = 2 \times \frac{25}{36} - 1 = \frac{50}{36} - 1 = \frac{14}{36} = \frac{7}{18}. \]

Step 4: Use Half-Angle Formulas

For half-angles: \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \) and \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \).

Step 5: Find \( \sin \frac{\theta}{2} \)

Substitute \( \cos \theta = \frac{5}{6} \) in the half-angle formula for sine: \[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{5}{6}}{2}} = \sqrt{\frac{\frac{1}{6}}{2}} = \sqrt{\frac{1}{12}} = \frac{\sqrt{3}}{6}. \]

Step 6: Find \( \cos \frac{\theta}{2} \)

Substitute \( \cos \theta = \frac{5}{6} \) in the half-angle formula for cosine: \[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{5}{6}}{2}} = \sqrt{\frac{\frac{11}{6}}{2}} = \sqrt{\frac{11}{12}} = \frac{\sqrt{33}}{6}. \]

Key Concepts

Double Angle Formulas

The double angle formulas are essential in trigonometry for transforming single angle expressions into double angle ones. They are particularly useful when dealing with problems requiring simplification or calculation of trigonometric expressions at double the angle.

• Formula for Sine: The double angle formula for sine is given by: \( \sin 2\theta = 2 \sin \theta \cos \theta \). This formula helps us find the sine of double the angle when we know the sine and cosine of the original angle.
• Formula for Cosine: The double angle formula for cosine is given by: \( \cos 2\theta = 2 \cos^2 \theta - 1 \) or alternatively \( \cos 2\theta = 1 - 2 \sin^2 \theta \). Depending on the given information, you can use either form to calculate the cosine of double the angle.

In practical applications like the original exercise where \( \cos \theta = \frac{5}{6} \), these formulas allow for effective computation of \( \sin 2\theta \) and \( \cos 2\theta \) by substituting known values and performing straightforward arithmetic.

Half-Angle Formulas

The half-angle formulas are another valuable tool in trigonometry. They allow us to express the trigonometric functions of half an angle in terms of the cosine (or sine) of the original angle itself. This is particularly useful when the angle is difficult to work with directly.

• Formula for Sine: The half-angle formula for sine is: \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \). This formula simplifies finding the sine of half an angle when \( \cos \theta \) is known.
• Formula for Cosine: The half-angle formula for cosine is: \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). Similarly, this helps you find the cosine of half an angle.

Using these formulas in the example with \( \cos \theta = \frac{5}{6} \), we can directly calculate the values of \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \) by substituting and solving the expressions.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone identities in trigonometry. It states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) equals one: \( \sin^2 \theta + \cos^2 \theta = 1 \).

• This identity is used to find the value of one trigonometric function given another. For instance, if you know \( \cos \theta \), you can calculate \( \sin \theta \) with this identity.
• It forms the basis for many other trigonometric identities and equations.

In the given exercise, since \( \cos \theta = \frac{5}{6} \), we apply this identity to find \( \sin \theta \) by solving \( \sin^2 \theta + (\frac{5}{6})^2 = 1 \). This understanding is vital for proceeding with calculations involving double and half-angle formulas where \( \sin \theta \) is needed.