Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 1
Given that \(\cos t=-\frac{2}{5}\) and that \(P(t)\) is a point in the second quadrant, find sin \(t\).
Short Answer
Expert verified
\( \sin t = \frac{\sqrt{21}}{5} \)
Step by step solution
01
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.
02
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 \( \sin^2 t \).
03
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:\[\sin^2 t + \frac{4}{25} = 1\]
04
Solving for Sine Squared
Subtract \( \frac{4}{25} \) from 1:\[\sin^2 t = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}\]
05
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:\[\sin t = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}\]
06
Conclusion
Given the calculations and the properties of the trigonometric functions in the second quadrant, we confirm that \( \sin t = \frac{\sqrt{21}}{5} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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掳 to 180掳, 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.
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.
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: \(\sin^2 t = \frac{21}{25}\).
