Let (theta) be an angle in quadrant II such that csc(theta)=8/5 Find the exact value of tan(theta) and cos(theta).
To find the exact value of tan(theta) and cos(theta), we can use the following relationships:
csc(theta) = 1/sin(theta)
tan(theta) = sin(theta)/cos(theta)
1 + tan^2(theta) = sec^2(theta)
First, we can find the value of sin(theta) using the given value of csc(theta):
csc(theta) = 8/5
Since csc(theta) = 1/sin(theta), we have:
1/sin(theta) = 8/5
Cross-multiplying, we get:
5 = 8sin(theta)
Dividing both sides by 8, we find:
sin(theta) = 5/8
Since theta is in quadrant II, we know that sin(theta) is positive.
Next, we can find the value of cos(theta) using the Pythagorean Identity:
sin^2(theta) + cos^2(theta) = 1
Using the value of sin(theta) we found, we have:
(5/8)^2 + cos^2(theta) = 1
Simplifying, we get:
25/64 + cos^2(theta) = 1
cos^2(theta) = 64/64 - 25/64
cos^2(theta) = 39/64
Taking the square root of both sides, we find:
cos(theta) = √(39/64)
Since theta is in quadrant II, we know that cos(theta) is negative.
Finally, we can find the value of tan(theta) using the relationship:
tan(theta) = sin(theta)/cos(theta)
Substituting the values we found, we have:
tan(theta) = (5/8) / (-√(39/64))
To simplify this, we can rationalize the denominator:
tan(theta) = (5/8) * (-√(64/39)) / (√(64/39))
tan(theta) = (5/8) * (-8/√39) * (√39/8)
tan(theta) = -5/39
In conclusion, the exact value of tan(theta) is -5/39 and the exact value of cos(theta) is -√(39/64).
To find the exact value of tan(theta) and cos(theta) when csc(theta) is given, we can use the following trigonometric identities:
csc(theta) = 1/sin(theta)
tan(theta) = sin(theta)/cos(theta)
Given that csc(theta) = 8/5, we can find sin(theta) by taking the reciprocal of csc(theta):
sin(theta) = 1/csc(theta) = 1/(8/5) = 5/8
Since we are given that theta is in quadrant II, where sine is positive, we know that sin(theta) = 5/8.
To find cos(theta), we can use the Pythagorean identity:
cos^2(theta) = 1 - sin^2(theta)
Plugging in the value of sin(theta), we have:
cos^2(theta) = 1 - (5/8)^2
cos^2(theta) = 1 - 25/64
cos^2(theta) = 64/64 - 25/64
cos^2(theta) = 39/64
Taking the square root of both sides, we get:
cos(theta) = ± sqrt(39/64)
Since theta is in quadrant II, where cosine is negative, we take the negative square root:
cos(theta) = -sqrt(39)/8
Therefore, the exact value of tan(theta) is sin(theta)/cos(theta), which is (5/8) / (-sqrt(39)/8), simplifying to:
tan(theta) = -5 / sqrt(39)
if csc Ø = 8/5
then
sin Ø = 5/8
I see a right-angled triangle with hypotenuse 8 and vertical side (the y) of 5
By Pythagoras, the other side is √39
in II both cosine and tangent are negative, so
tanØ = -5/√39
cosØ = -√39/8