QuestionAugust 27, 2025

Theta =(-7pi )/(4) then find exact values for the following: sec(Theta ) equals square csc(Theta ) equals square tan(Theta ) equals square cot(Theta ) equals square

Theta =(-7pi )/(4) then find exact values for the following: sec(Theta ) equals square csc(Theta ) equals square tan(Theta ) equals square cot(Theta ) equals square
Theta =(-7pi )/(4) then find exact values for the following: sec(Theta ) equals square csc(Theta ) equals square tan(Theta ) equals square cot(Theta ) equals square

Solution
3.8(262 votes)

Answer

sec(\Theta) = \sqrt{2} ### csc(\Theta) = \sqrt{2} ### tan(\Theta) = 1 ### cot(\Theta) = 1 Explanation 1. Determine the reference angle The angle \Theta = \frac{-7\pi}{4} is negative. Add 2\pi to find a positive coterminal angle: \Theta + 2\pi = \frac{-7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}. 2. Calculate sec(\Theta) For \Theta = \frac{\pi}{4}, sec(\Theta) = \frac{1}{cos(\Theta)}. Since cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, sec(\Theta) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}. 3. Calculate csc(\Theta) For \Theta = \frac{\pi}{4}, csc(\Theta) = \frac{1}{sin(\Theta)}. Since sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, csc(\Theta) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}. 4. Calculate tan(\Theta) For \Theta = \frac{\pi}{4}, tan(\Theta) = \frac{sin(\Theta)}{cos(\Theta)}. Since both sin(\frac{\pi}{4}) and cos(\frac{\pi}{4}) are \frac{\sqrt{2}}{2}, tan(\Theta) = 1. 5. Calculate cot(\Theta) For \Theta = \frac{\pi}{4}, cot(\Theta) = \frac{1}{tan(\Theta)}. Since tan(\Theta) = 1, cot(\Theta) = 1.

Explanation

1. Determine the reference angle<br /> The angle $\Theta = \frac{-7\pi}{4}$ is negative. Add $2\pi$ to find a positive coterminal angle: $\Theta + 2\pi = \frac{-7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}$.<br /><br />2. Calculate $sec(\Theta)$<br /> For $\Theta = \frac{\pi}{4}$, $sec(\Theta) = \frac{1}{cos(\Theta)}$. Since $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $sec(\Theta) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$.<br /><br />3. Calculate $csc(\Theta)$<br /> For $\Theta = \frac{\pi}{4}$, $csc(\Theta) = \frac{1}{sin(\Theta)}$. Since $sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $csc(\Theta) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$.<br /><br />4. Calculate $tan(\Theta)$<br /> For $\Theta = \frac{\pi}{4}$, $tan(\Theta) = \frac{sin(\Theta)}{cos(\Theta)}$. Since both $sin(\frac{\pi}{4})$ and $cos(\frac{\pi}{4})$ are $\frac{\sqrt{2}}{2}$, $tan(\Theta) = 1$.<br /><br />5. Calculate $cot(\Theta)$<br /> For $\Theta = \frac{\pi}{4}$, $cot(\Theta) = \frac{1}{tan(\Theta)}$. Since $tan(\Theta) = 1$, $cot(\Theta) = 1$.
Click to rate:

Similar Questions