Sin theta=8/17, theta in Quadrant I, Cos X=-sqrt5/5 in Quadrant II
I was expecting some kind of question.
It says to evaluate the expression
"to evaluate the expression" makes no sense
we can find the angles.
Ø = appr 28°
X = appr 116.6°
To find the value of theta in Quadrant I, where sin theta = 8/17, you can follow these steps:
1. Start by drawing a right triangle in Quadrant I. Let's label the sides as follows:
- The side opposite to theta as "opp" (length unknown).
- The side adjacent to theta as "adj" (length unknown).
- The hypotenuse as "hyp" (length = 17 since sin theta = 8/17).
2. Use the Pythagorean theorem to find the length of the remaining side:
- hyp^2 = opp^2 + adj^2
- 17^2 = opp^2 + adj^2
- 289 = opp^2 + adj^2
3. Since sin theta = opp/hyp, substitute the given value for sin theta:
- 8/17 = opp/17
4. Solve for the value of opp:
- Cross-multiplying, 17 * (8/17) = opp
- 8 = opp
5. Now, we know that the length of the opposite side (opp) is 8.
6. To find the length of the adjacent side (adj), use the Pythagorean theorem again:
- 289 = 8^2 + adj^2
- 289 = 64 + adj^2
- adj^2 = 225
- adj = 15 (since theta is in Quadrant I, the adjacent side length is positive)
7. Now, you have the values of opp (8) and adj (15). Use the trigonometric identity "cos theta = adj/hyp" to find the value of cos theta:
- cos theta = 15/17 (since hyp = 17)
Therefore, in Quadrant I, where sin theta = 8/17, the value of cos theta is 15/17.
Moving on to the second part of the question, to find theta in Quadrant II where cos x = -√5/5, you can proceed with the following steps:
1. Start by drawing a right triangle in Quadrant II. Let's label the sides as follows:
- The side opposite to x as "opp" (length unknown).
- The side adjacent to x as "adj" (length unknown).
- The hypotenuse as "hyp" (length = 5 since cos x = -√5/5).
2. Use the Pythagorean theorem to find the length of the remaining side:
- hyp^2 = opp^2 + adj^2
- 5^2 = opp^2 + adj^2
- 25 = opp^2 + adj^2
3. Since cos x = adj/hyp, substitute the given value for cos x:
- -√5/5 = adj/5
4. Solve for the value of adj:
- Cross-multiplying, 5 * (-√5/5) = adj
- -√5 = adj
5. Now, we know that the length of the adjacent side (adj) is -√5.
(Note: In Quadrant II, the adjacent side length is negative)
6. To find the length of the opposite side (opp), use the Pythagorean theorem again:
- 25 = opp^2 + (-√5)^2
- 25 = opp^2 + 5
- opp^2 = 20
- opp = √20 or 2√5
7. Now, you have the values of adj (-√5) and opp (2√5). Use the trigonometric identity "sin x = opp/hyp" to find the value of sin x:
- sin x = (2√5)/5 (since hyp = 5)
Therefore, in Quadrant II, where cos x = -√5/5, the value of sin x is (2√5)/5.