Problem 1 - Precalculus with Calculus Previews

Question

Given that \(\cos t=-\frac{2}{5}\) and that \(P(t)\) is a point in the second quadrant, find sin \(t\).

Step-by-Step Solution

Answer

\( \sin t = \frac{\sqrt{21}}{5} \)

Step 1: Understanding the Problem

We are given that \( \cos t = -\frac{2}{5} \) and the angle \( t \) is located in the second quadrant. We need to find \( \sin t \). Since \( t \) is in the second quadrant, the cosine of \( t \) is negative, and the sine of \( t \) is positive.

Step 2: Using the Pythagorean Identity

We use the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \cos t = -\frac{2}{5} \) into the identity to find \( \sin^2 t \).

Step 3: Plugging Cosine into Identity

Plug \( \cos t = -\frac{2}{5} \) into the identity:\[\sin^2 t + \left(-\frac{2}{5}\right)^2 = 1\]This simplifies to:\[\sin^2 t + \frac{4}{25} = 1\]

Step 4: Solving for Sine Squared

Subtract \( \frac{4}{25} \) from 1:\[\sin^2 t = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}\]

Step 5: Finding Sine

Taking the square root of both sides gives:\[\sin t = \pm \sqrt{\frac{21}{25}}\]Since \( t \) is in the second quadrant where sine is positive, we choose the positive root:\[\sin t = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}\]

Step 6: Conclusion

Given the calculations and the properties of the trigonometric functions in the second quadrant, we confirm that \( \sin t = \frac{\sqrt{21}}{5} \).

Key Concepts

Second Quadrant

The second quadrant of a coordinate plane is a fascinating area where specific properties of trigonometric functions emerge. It lies between the angles of 90° to 180°, or \(\frac{\pi}{2}\) to \(\pi\) radians. In this quadrant:

• The cosine function is negative because it represents the x-coordinate, which moves to the left of the origin.
• The sine function remains positive as the y-coordinate value extends upward, above the origin.

Knowing these sign rules is crucial when solving problems involving trigonometric identities because they help determine the sign of the computed values.

Pythagorean Identity

The Pythagorean identity is one of trigonometry's cornerstones. It states that for any angle \(t\):\[\sin^2 t + \cos^2 t = 1\]This identity mirrors the Pythagorean theorem, showing how the lengths of the sides of a right triangle relate on a unit circle. By plugging in the known cosine value, you can solve for the sine value, which is exactly what we do here: substitute \(\cos t = -\frac{2}{5}\) into the equation. This simplifies the problem down and allows for solving \(\sin^2 t\). Understanding this identity is essential to mastering trigonometry.

Cosine Function

The cosine function is deeply tied to the x-axis of the coordinate plane. Its values tell us how far left or right a point is on the unit circle:

• In the second quadrant, cosine values are always negative because it relates to the x-coordinate being negative.
• The given value, \(\cos t = -\frac{2}{5}\), tells us that the point is 2/5 units to the left of the circle's radius.

Cosine values play a critical role in certain mathematical identities, like the Pythagorean identity, which we utilize to find the sine function.

Sine Function

The sine function represents the y-axis component on a unit circle. It measures how far above or below the point is compared to the origin.

• In the second quadrant, the sine of an angle is positive due to it's relation to the upward direction of the y-axis.
• When cosine is known, the sine value can be derived using identities such as the Pythagorean identity: \(\sin^2 t = \frac{21}{25}\).

Taking the square root of \(\sin^2 t\) gives the final sine value. In our example, \(\sin t = \frac{\sqrt{21}}{5}\). Understanding the behavior of the sine function in different quadrants is fundamental to comprehending trigonometric equations.