Angle [tex]\theta[/tex] is in standard position. If [tex]sin (\theta)=-\frac{1}{3}[/tex], and [tex]\pi\ \textless \ \theta\ \textless \ \frac{3 \pi}{2}[/tex], find [tex]cos (\theta)[/tex].
A. [tex]-\frac{2 \sqrt{2}}{3}[/tex]
B. [tex]-\frac{4}{3}[/tex]
C. [tex]\frac{4}{3}[/tex]
D. [tex]\frac{2 \sqrt{2}}{3}[/tex]
Answer (1)
Use the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 .
Substitute the given value sin ( θ ) = − 3 1 into the identity and solve for cos 2 ( θ ) .
Take the square root to find cos ( θ ) = ± 3 2 2 .
Determine the sign of cos ( θ ) based on the quadrant of θ , which is the third quadrant where cosine is negative, so cos ( θ ) = − 3 2 2 .
− 3 2 2
Explanation
Analyze the given information We are given that sin ( θ ) = − 3 1 and π < θ < 2 3 π . We need to find cos ( θ ) . The interval π < θ < 2 3 π corresponds to the third quadrant in the coordinate plane. In the third quadrant, both sine and cosine are negative.
Apply the Pythagorean identity We can use the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 to find cos ( θ ) . Substituting the given value of sin ( θ ) into the identity, we get:
( − 3 1 ) 2 + cos 2 ( θ ) = 1
9 1 + cos 2 ( θ ) = 1
cos 2 ( θ ) = 1 − 9 1
cos 2 ( θ ) = 9 8
Solve for cos(theta) Taking the square root of both sides, we have:
cos ( θ ) = ± 9 8 = ± 9 8 = ± 3 2 2
Determine the sign of cos(theta) Since θ is in the third quadrant, where cosine is negative, we choose the negative value:
cos ( θ ) = − 3 2 2
Final Answer Therefore, the value of cos ( θ ) is − 3 2 2 .
Examples
Understanding trigonometric functions and their values in different quadrants is crucial in various fields such as physics and engineering. For example, when analyzing the motion of a pendulum, knowing the sine and cosine of the angle helps determine the pendulum's position and velocity at any given time. Similarly, in electrical engineering, these functions are used to describe alternating current (AC) circuits, where the voltage and current vary sinusoidally with time. Thus, mastering these concepts provides a foundation for solving real-world problems in these disciplines.
