Three trigonometric functions for a given angle are shown below.
[tex]$\sin \theta=-\frac{77}{85}, \cos \theta=\frac{36}{85}, \tan \theta=-\frac{77}{36}$[/tex]
What are the coordinates of point ([tex]$x, y$[/tex]) on the terminal ray of angle [tex]$\theta$[/tex], assuming that the values above were not simplified?
A. ([tex]$-77,-36$[/tex])
B. ([tex]$-77,36$[/tex])
C. ([tex]$-36,77$[/tex])
D. ([tex]$36,-77$[/tex])
Answer (2)
The problem provides the values of sin θ , cos θ , and tan θ .
We use the definitions sin θ = r y , cos θ = r x , and tan θ = x y to relate the trigonometric functions to the coordinates ( x , y ) of a point on the terminal ray of angle θ .
By comparing the given values with the definitions, we find x = 36 and y = − 77 .
The coordinates of the point are ( 36 , − 77 ) .
Explanation
Analyze the given information. We are given the following trigonometric functions for an angle θ :
sin θ = − 85 77 , cos θ = 85 36 , tan θ = − 36 77 We need to find the coordinates ( x , y ) of a point on the terminal ray of angle θ , assuming the values above were not simplified.
Recall trigonometric definitions. Recall the definitions of trigonometric functions in terms of coordinates on the unit circle: sin θ = r y , cos θ = r x , tan θ = x y where r is the distance from the origin to the point ( x , y ) , and 0"> r > 0 .
Determine the coordinates. From the given information, we have: sin θ = − 85 77 = r y cos θ = 85 36 = r x Since the values are not simplified, we can directly read off the values of x , y , and r (up to a scaling factor). We can take r = 85 , which gives us y = − 77 and x = 36 . Thus, the coordinates of the point are ( 36 , − 77 ) .
We can verify this with the tangent function: tan θ = x y = 36 − 77 = − 36 77 This matches the given value of tan θ .
State the final answer. Therefore, the coordinates of the point ( x , y ) on the terminal ray of angle θ are ( 36 , − 77 ) .
Examples
Understanding trigonometric functions and their relationship to coordinates on a circle is crucial in various fields. For example, in physics, when analyzing projectile motion, we use trigonometric functions to break down the initial velocity into horizontal and vertical components. If a projectile is launched with an initial velocity v at an angle θ to the horizontal, the horizontal component is v cos θ and the vertical component is v sin θ . Knowing these components helps us predict the range and maximum height of the projectile.
The coordinates of the point (x, y) on the terminal ray of angle θ are (36, -77), derived from the given sine and cosine values. This matches the definitions of trigonometric functions in relation to coordinates on a circle. Therefore, the answer is option D: (36, -77).
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