Question

three trigonometric functions for a given angle are shown below.\\(\sin \theta = -\frac{77}{85}\\), \\(\cos \theta = \frac{36}{85}\\), \\(\tan \theta = -\frac{77}{36}\\)\
what are the coordinates of point \\((x, y)\\) on the terminal ray of angle \\(\theta\\), assuming that the values above were not simplified?\
\\(\circ\\) \\((-77, -36)\\)\
\\(\circ\\) \\((-77, 36)\\)\
\\(\circ\\) \\((-36, 77)\\)\
\\(\circ\\) \\((36, -77)\\)

Explanation:

Step1: Recall trigonometric definitions

For a point \((x, y)\) on the terminal ray of angle \(\theta\) and \(r=\sqrt{x^{2}+y^{2}}\), we have \(\sin\theta=\frac{y}{r}\), \(\cos\theta = \frac{x}{r}\), and \(\tan\theta=\frac{y}{x}\) (where \(x
eq0\)).

Step2: Identify \(x\), \(y\) from given trigonometric values

We are given \(\sin\theta=-\frac{77}{85}\), \(\cos\theta=\frac{36}{85}\), \(\tan\theta = -\frac{77}{36}\). From \(\sin\theta=\frac{y}{r}\) and \(\cos\theta=\frac{x}{r}\), if we assume \(r = 85\) (since the values are not simplified, we can take \(r\) as the denominator), then \(y=\sin\theta\times r=-\frac{77}{85}\times85=- 77\) and \(x=\cos\theta\times r=\frac{36}{85}\times85 = 36\). So the coordinates \((x,y)\) should be \((36,-77)\).

Answer:

D. \((36, -77)\) (assuming the options are labeled as A. \((-77, -36)\), B. \((-77, 36)\), C. \((-36, 77)\), D. \((36, -77)\))