91Ó°ÊÓ

Half-angle identities are essential in trigonometry and help us find the values of trigonometric functions for half an angle using the known values of the angle itself. It simplifies the process when exact values are required.
For instance, the half-angle identity for sine is given by:
\(\text{sin} \frac{\theta}{2} = \pm \text{sqrt}\frac{1 - \text{cos}\theta}{2} \)
The ‘plus minus’ sign depends on the quadrant where the half-angle falls. Understanding the unit circle helps determine this.
Similarly, the half-angle identity for cosine is:
\(\text{cos} \frac{\theta}{2} = \pm \text{sqrt}\frac{1 + \text{cos}\theta}{2} \)

• If the half-angle is in the first or fourth quadrant, the sine and cosine values are positive.
• If it is in the second or third quadrant, they are negative.

In the problem, \( \theta \) was in the second quadrant, so \(\text{cos} \theta < 0 \) and \( \frac{\theta}{2} \) falls in the first quadrant where trigonometric values are positive.

Double-angle identities are useful to express trigonometric functions of an angle that is twice another angle. These identities make it easier to solve many trigonometric equations.
For sine, the double-angle identity is:
\(\text{sin}(2\theta) = 2 \text{sin}\theta \text{cos}\theta \) We used this identity directly in the solution by substituting the values of \( \text{sin} \theta = -\frac{1}{\text{sqrt}(10)}\) and \( \text{cos} \theta = -\frac{3}{\text{sqrt}(10)}\).

For cosine, the double-angle identity is:
\(\text{cos}(2\theta) = \text{cos}^2\theta - \text{sin}^2\theta \)
This identity was used in the solution by substituting the same values of \( \text{sin} \theta \) and \( \text{cos} \theta \). These identities come from the fundamental angle sum identities.
Doubling the angle helps connect and evaluate trigonometric functions at doubled angles.

The Pythagorean identity establishes a fundamental relationship among the sine, cosine, and tangent functions. This identity is especially useful for solving problems where we have one trigonometric ratio and need to find the others.
The primary Pythagorean identity is:
\(\text{sin}^2\theta + \text{cos}^2\theta = 1 \)
This is derived from the Pythagorean theorem when applied to the unit circle.
In the exercise, knowing \( \text{tan} \theta \) and \( \text{cos} \theta < 0 \), we used the identity to solve for \( \text{cos} \theta \) and \( \text{sin} \theta \):

• Express \( \text{sin} \theta \) in terms of \( \text{cos} \theta \).
• Substitute into the Pythagorean identity to solve for \( \text{cos} \theta \).
• Derive \( \text{sin} \theta \) from the known value of \( \text{cos} \theta \).

Utilizing this identity allows us to transition between different trigonometric functions seamlessly.

Each of the four quadrants in the unit circle corresponds to different signs for the trigonometric functions sine, cosine, and tangent.

• The first quadrant \( (0 \text{to} \frac{\pi}{2}) \) has positive sine and cosine values.
• The second quadrant \( (\frac{\pi}{2} \text{to} \pi) \) has positive sine but negative cosine values.
• The third quadrant \( (\frac{\text{\pi}}{2} \text{to} \frac{3\text{\pi}}{2}) \) has negative sine and cosine values.
• The fourth quadrant \( (\frac{3\text{\pi}}{2} \text{to} 2\text{\pi}) \) has negative sine but positive cosine values.

Understanding which quadrant an angle lies in helps determine the signs of its trigonometric functions.
In our solution, knowing \( \cos \theta < 0 \) placed \( \theta \) in the second or third quadrant. By using the tangent condition, we narrowed it down further to determine the exact function values correctly.