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.
- I need help getting started in the right direction. Please be specific as I am new to trig.
for a triangle in standard position, sinC = y/r
since r is always positive, you must have y>0. That is QI and QII.
Since cosC = x/r, you must have x<0, so QII
You have a 7-24-25 triangle, so the terminal point of the hypotenuse lies at (-7,24).
If you are new to this, just recall that it is almost always a good place to start by drawing the triangle in standard position, and recalling that
sinC = y/r
cosC = x/r
tanC = y/x
That will get you 90% of the way there every time.
Sure! Let's break down the problem step by step.
a) To determine the possible quadrants in which ∠C may lie, we need to use the given information sin C = 7/25. Recall that sine is positive in the first and second quadrants. We also know that sin C = opposite/hypotenuse, where opposite is 7 and hypotenuse is 25. By using the Pythagorean theorem, we can find the adjacent side. Let's call it x.
Using the Pythagorean theorem: x² + 7² = 25²
=> x² + 49 = 625
=> x² = 576
=> x = √576
=> x = 24 or -24
Since sin C is positive, we can eliminate the possibility of -24. Therefore, x = 24, which means that the adjacent side is positive. With this information, we can conclude that ∠C may lie in the first quadrant.
b) If we know that cos C is negative, it means that the adjacent side is negative. In this case, from part a), we have determined that the adjacent side is positive. So, if cos C is negative, we need to change our answer from part a). Therefore, if cos C is negative, ∠C may lie in the second quadrant.
c) To sketch a diagram representing ∠C in standard position, we start with the positive x-axis horizontal and the positive y-axis vertical. In the first quadrant, the terminal arm lies in the upper-right direction. Since cos C is negative, according to part b), we need to adjust the terminal arm to the second quadrant, which is upper-left. A sketch of ∠C in standard position in the second quadrant would look something like this:
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d) To find the coordinates of a point P on the terminal arm of ∠C, we need to use the values of the trigonometric ratios. We know that sin C = 7/25 and cos C is negative. In the second quadrant, both sin C and cos C are negative, which means the point P will have negative x and y coordinates.
Let's call the coordinates of P as (x, y). Since sin C = opposite/hypotenuse = 7/25, we get y/25 = -7/25 => y = -7.
Similarly, since cos C = adjacent/hypotenuse, we can use the Pythagorean theorem to find the value of x. From part a), we found that the adjacent side is positive, and since cos C is negative, we need to make it negative. Therefore, x = -24.
So, the coordinates of point P are (-24, -7).
e) To find the other two primary trigonometric ratios, we can use the coordinates of point P from part d).
The other two primary trigonometric ratios are:
- Tan C = opposite/adjacent = -7/-24 = 7/24
- Cot C = adjacent/opposite = -24/-7 = 24/7
I hope this helps you get started with trigonometry! Let me know if you have any further questions.