Consider ∠C such that sin C = 7/25
a) What are the possible quadrants in which ∠C may lie?
b) If you know that cos C is negative, how does your answer to part a) change?
c) Sketch a diagram to represent ∠C in standard position, given that the condition in part b) is true.
d) Find coordinates of a point P on the terminal arm of ∠C.
e) Write exact expressions for the other two primary trigonometric ratios for ∠C.
a) sin is + in I and II
b) cos is - in II so there only
c) hard to sketch here from origin up 7 distance 25
x^2 + 7^2 =625 and x is -
x^2 = 625 - 49 = 576
x = -24
so point is at (-24,7)
cos C = -24/25
tan C = -7/24
a) To determine the possible quadrants in which ∠C may lie, we need to consider the sign of the sine function. Since sin C = 7/25, which is positive, we know that ∠C can lie in quadrants I and II.
b) If we know that cos C is negative, it means that ∠C is in quadrant II. Therefore, our answer to part a) changes from the possible quadrants being I and II to only II.
c) In quadrant II, the angle ∠C is between 90° and 180°. We can represent this angle in standard position by drawing the positive x-axis and the positive y-axis, intersecting at the origin. Then, we measure an angle of less than 180° in the counter-clockwise direction from the positive x-axis. This will represent ∠C in standard position.
d) To find the coordinates of a point P on the terminal arm of ∠C, we can use the given information sin C = 7/25. By remembering that sin C is the ratio of the length of the side opposite to the angle C to the hypotenuse, we can set up a right triangle with the side opposite to C equal to 7 and the hypotenuse equal to 25. Using the Pythagorean theorem, we can find the missing side:
a^2 + b^2 = c^2
(7)^2 + b^2 = (25)^2
49 + b^2 = 625
b^2 = 625 - 49
b^2 = 576
b = 24
Therefore, the coordinates of point P on the terminal arm of ∠C are (x, y) = (-25, 24).
e) The other two primary trigonometric ratios for ∠C can be found using the coordinates of point P. The cosine (cos) of ∠C is defined as the ratio of the length of the side adjacent to ∠C to the hypotenuse. In this case, the side adjacent to ∠C is -25 (negative because it's in quadrant II) and the hypotenuse is 25. Therefore, cos C = -25/25 = -1.
The tangent (tan) of ∠C is defined as the ratio of the length of the side opposite to ∠C to the side adjacent to ∠C. In this case, the side opposite to ∠C is 7 and the side adjacent to ∠C is -25. Therefore, tan C = 7/-25 = -7/25.
To summarize:
- cos C = -1
- tan C = -7/25