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Chapter 4: Problem 26

Evaluate (if possible) the six trigonometric functions at the real number. $$t=7 \pi / 4$$

Short Answer

Expert verified
The six trigonometric functions at \( t = \frac{7 \pi }{4} \) are \( \sin(t) = -\sqrt{2}/2 \), \( \cos(t) = \sqrt{2}/2 \), \( \tan(t) = - 1 \), \( \cot(t) = - 1 \), \( \sec(t) = \sqrt{2} \), and \( \csc(t) = -\sqrt{2} \).

Step by step solution

01

Convert the Given Value into Degrees

The given value \( t = \frac{7 \pi }{4} \) is in radians. The conversion \( t = \frac{180 \times t }{\pi } \) will convert this value into degrees, which will result in \( t = 315 \) degrees.
02

Use Unit Circle to find Sin(x) and Cos(x) values

Now, look at the unit circle. The coordinates on the unit circle represent the [cos(x), sin(x)] at any given x. In this case, for \( t = 315 \) degrees, we find the coordinates (cos(x), sin(x)) to be \(\sqrt{2}/2, -\sqrt{2}/2\). So, \( \sin(t) = -\sqrt{2}/2 \) and \( \cos(t) = \sqrt{2}/2 \).
03

Using Values of Sin(x) and Cos(x) to find other functions

Knowing the sine and cosine values, we can calculate tangent, cotangent, secant, and cosecant. \( \tan(t)= \sin(t)/ \cos(t) = (-\sqrt{2}/2) / (\sqrt{2}/2) = -1 \), \( \cot(t)= 1/\tan(t)= -1 \), \( \sec(t) = 1/ \cos(t)= 2/\sqrt{2}= \sqrt{2} \), \( \csc(t) = 1/ \sin(t) = -2/\sqrt{2}=-\sqrt{2} \).

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