Find the remaining five trigonometric functions of [tex]\theta[/tex].
[tex]\sin \theta=\frac{2}{3}, \theta \text { in quadrant II }[/tex]
Complete the following table.
[tex]\begin{array}{ll}<
\sin \theta=\frac{2}{3} & \csc \theta=\square \\
\cos \theta=\square & \sec \theta=\square \\
\tan \theta=\square & \cot \theta=\square
\end{array}[/tex]
Answer (2)
Find csc θ using the reciprocal identity: csc θ = s i n θ 1 = 2 3 .
Find cos θ using the Pythagorean identity and quadrant information: cos θ = − 3 5 .
Find tan θ using the quotient identity: tan θ = c o s θ s i n θ = − 5 2 5 .
Find cot θ and sec θ using reciprocal identities: cot θ = − 2 5 and sec θ = − 5 3 5 .
Explanation
Problem Analysis We are given that sin θ = 3 2 and that θ is in quadrant II. Our goal is to find the values of the remaining five trigonometric functions: cos θ , tan θ , csc θ , sec θ , and cot θ .
Finding csc θ First, we can find csc θ using the identity csc θ = s i n θ 1 . Since sin θ = 3 2 , we have csc θ = 3 2 1 = 2 3
Finding cos θ Next, we can find cos θ using the Pythagorean identity sin 2 θ + cos 2 θ = 1 . We have cos 2 θ = 1 − sin 2 θ = 1 − ( 3 2 ) 2 = 1 − 9 4 = 9 5 Taking the square root of both sides, we get cos θ = ± 9 5 = ± 3 5 . Since θ is in quadrant II, cos θ is negative. Therefore, cos θ = − 3 5
Finding tan θ Now, we can find tan θ using the identity tan θ = c o s θ s i n θ . We have tan θ = − 3 5 3 2 = 3 2 ⋅ − 5 3 = − 5 2 = − 5 2 5
Finding cot θ We can find cot θ using the identity cot θ = t a n θ 1 . We have cot θ = − 5 2 5 1 = − 2 5 5 = − 10 5 5 = − 2 5
Finding sec θ Finally, we can find sec θ using the identity sec θ = c o s θ 1 . We have sec θ = − 3 5 1 = − 5 3 = − 5 3 5
Final Answer In summary, we have: sin θ = 3 2 cos θ = − 3 5 tan θ = − 5 2 5 csc θ = 2 3 sec θ = − 5 3 5 cot θ = − 2 5
Examples
Trigonometric functions are essential in various fields, including physics, engineering, and navigation. For instance, when analyzing the motion of a pendulum, the angle it makes with the vertical changes over time, and this angle can be described using trigonometric functions. Similarly, in electrical engineering, alternating current (AC) waveforms are often modeled using sine and cosine functions. Understanding how to find the values of these functions for different angles and quadrants is crucial for solving real-world problems in these areas.
The remaining five trigonometric functions for θ where sin θ = 3 2 in quadrant II are: csc θ = 2 3 , cos θ = − 3 5 , sec θ = − 5 3 5 , tan θ = − 5 2 5 , and cot θ = − 2 5 .
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