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Chapter 4: Problem 2

Find all numbers \(t\) such that \(\left(\frac{3}{5}, t\right)\) is a point on the unit circle.

Short Answer

Expert verified
The possible values of \(t\) that make \(\left(\frac{3}{5},t\right)\) a point on the unit circle are \(t = \frac{4}{5}\) or \(t = -\frac{4}{5}\).

Step by step solution

01

Write down the equation of the unit circle with the given x-coordinate

We know that points on the unit circle satisfy the equation \(x^2 + y^2 = 1\). Replace x with the given x-coordinate, \(\frac{3}{5}\), to get the equation: \[\left(\frac{3}{5}\right)^2 + t^2 = 1\]
02

Solve for the possible values of t

Next, we will solve the equation to find the possible values of the y-coordinate (t). First, square \(\frac{3}{5}\) and simplify: \[\left(\frac{3}{5}\right)^2 = \frac{9}{25}\] Now, we can substitute this value back into the equation and solve for t: \(\frac{9}{25} + t^2 = 1\) To find the possible values of t, first subtract \(\frac{9}{25}\) from both sides of the equation: \(t^2 = 1 - \frac{9}{25}\) Now, simplify the right side of the equation: \(t^2 = \frac{16}{25}\) Lastly, take the square root of both sides: \(t = \pm\frac{4}{5}\)
03

State the solution

Thus, there are two possible values for the y-coordinate (t) that make \(\left(\frac{3}{5},t\right)\) a point on the unit circle: \(t = \frac{4}{5}\) or \(t = -\frac{4}{5}\)

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In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\tan \frac{17 \pi}{8}\)

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