Consider angle C such that sin C= 7/25
Sketch a diagram to represent angle C in standard position if cos C is negative
Find the co-ordinates of a point P on the terminal arm of angle C.
You should have been told about the CAST rule, that is, an acronym telling you in which quadrants trig ratios are positive.
Since sinC is positive and cosC is negative
then C must be in quadrant II
since sinØ = opposite/hyptenuse
sinC = 7/25 --> y=7, r=25
x^2 + y^2= r^2
x^2 + 49 = 625
x^2 = 576
x = ±24, but we are in quad II, so x = -24
your terminal point is (-24,7)
join to the centre and finish the right-angled triangle
X^2 + Y^2 = C^2.
x^2 + 7^2 = 25^2,
x^2 = 25^2 - 7^2 = 576, x = 24.
If Cos C is negative, X is negative. Cos C = (-24)/25 = -0.96, C = 163.7o CCW from +x-axis. = 73.7o W. of N.
P(-24,7).
To sketch angle C in standard position, we will first start with the initial arm. The initial arm is the positive x-axis on the coordinate plane. Since cosine (cos) C is negative, angle C will lie in either the second or third quadrant.
Let's assume angle C lies in the second quadrant. To find the coordinates of point P on the terminal arm of angle C, we need to use the trigonometric definitions of sine and cosine.
Given that sin C = 7/25, we can find cos C using the Pythagorean identity sin^2 C + cos^2 C = 1.
sin^2 C + cos^2 C = 1
(7/25)^2 + cos^2 C = 1
49/625 + cos^2 C = 1
cos^2 C = 1 - 49/625
cos^2 C = (625 - 49)/625
cos^2 C = 576/625
cos C = √(576/625)
cos C = ± 24/25 (since cos C is negative)
So, we have cos C = -24/25.
Assuming angle C lies in the second quadrant, the x-coordinate of point P will be negative, and the y-coordinate will be positive to ensure sin C is positive (since sin C = 7/25).
For example, let's assume angle C is in the second quadrant and cos C = -24/25. Suppose we choose the radius to be 25 units (this is just an arbitrary choice). Since the x-coordinate is negative, we move 24 units to the left from the origin (0,0), and since the y-coordinate is positive, we move 7 units up. Therefore, one possible set of coordinates for point P is (-24, 7).
The diagram below represents angle C in the second quadrant with point P at coordinates (-24, 7):
|
|
|
-------+-------
P |
|
|
|
Note that if we assumed angle C to lie in the third quadrant, the x-coordinate would still be negative, but the y-coordinate would be negative to ensure sin C is positive. In that case, the coordinates of point P would be (-24, -7).