If cos theta = 0.8 and 270<theta<360, find the exact value of sin 2theta
a) -0.96
b) -0.6
c) 0.96
d) 0.28
Answer: b
2) If csc theta = -5/3, and theta has its terminal side in Quadrant III, find the exact value of tan 2theta
a)24/25
b) 7/25
c) 24/7
d) -7/25
Thanks
1) cosθ=0.8 , θ is in fourth quadrant
θ= cos^(-1)0.8=36.86989765 + 270=306.87
so , sin 2θ= sin2(306.87)= -0.96
answer : a
2) csc Ø = -5/3, then sin Ø = -3/5 and is in III
the cos Ø = -4/5
so tan 2Ø = sin 2Ø/cos 2Ø
= 2sinØcosØ/(cos^2 Ø - sin^2 Ø)
=2(-3/5)(-4/5)/(16/25 - 9/25)
= 24/7
1) cscθ = (-5/3) , θ is in third quadrant
1/sinθ = (-5/3) => sinθ= -3/5
θ= -36.8699
θ is in third quadrant
so θ = 36.8669+180 = 216.8669
tan2θ = tan2(216.8669) = 3.42724 = 24/7
answer : c
When a trig question asks for "exact" value, a calculator is not to be used, and all work has to be shown with exact values.
24/7 is not equal to 3.42724
thanks for note Reiny .
To find the exact value of sin 2theta, we need to use the double-angle formula for sine, which states that sin 2theta = 2sin theta cos theta.
Given that cos theta = 0.8, we can find sin theta using the Pythagorean identity. The Pythagorean identity states that sin^2 theta + cos^2 theta = 1. So, sin theta = sqrt(1 - cos^2 theta).
Since the angle theta is in the range 270<theta<360, which is in Quadrant IV, sin theta is negative. Therefore, sin theta = -sqrt(1 - cos^2 theta).
Substituting the values we have:
sin theta = -sqrt(1 - (0.8)^2) = -sqrt(1 - 0.64) = -sqrt(0.36) = -0.6
Now we can find sin 2theta:
sin 2theta = 2sin theta cos theta = 2(-0.6)(0.8) = -0.96
Therefore, the correct answer is (a) -0.96.
For the second question, to find the exact value of tan 2theta, we need to use the double-angle formula for tangent, which states that tan 2theta = (2tan theta)/(1 - tan^2 theta).
Given that csc theta = -5/3, we can find sin theta using the reciprocal identity. The reciprocal identity states that csc theta = 1/sin theta. Therefore, sin theta = 1/(csc theta) = -3/5.
Since the angle theta has its terminal side in Quadrant III, both sin theta and csc theta are negative.
Now we can find tan theta. The tangent of an angle can be found using the ratio of sin theta to cos theta. Since we already know sin theta, we need to find cos theta.
Using the Pythagorean identity, cos^2 theta = 1 - sin^2 theta = 1 - (-3/5)^2 = 1 - 9/25 = 16/25. Taking the square root of both sides, we get cos theta = sqrt(16/25) = 4/5.
Now we can find tan theta:
tan theta = sin theta / cos theta = (-3/5) / (4/5) = -3/4.
Using the double-angle formula for tangent, we can find tan 2theta:
tan 2theta = (2tan theta) / (1 - tan^2 theta) = (2(-3/4)) / (1 - (-3/4)^2) = (-6/4) / (1 - 9/16) = (-3/2) / (7/16) = -24/7.
Therefore, the correct answer is (c) -24/7.