Find the exact value of each expression if 0 degrees < theta < 90 degrees
If tan (theta) = 1/2 , find cos(theta)
If cos(theta) = 8/17, find csc (theta)
Find the exact value of each expression is 180 degrees < theta < 270 degrees
7. If tan (theta)= 1, find cos(theta)
Tangent is opposite / adjacent, so we can say that in tan (theta) = 1/2, the 1 and 2 are the sides of the right triangle and
c^2 = a^2 + b^2
c^2 = 1^2 + 2^2
c = sqrt(5) is the hypotenuse.
Cosine is adjecent / hypotenuse, so
cos(theta) = 2 / sqrt(5)
Method is the same for other questions.
1. If tan(theta) = 1/2, we can use the definition of tangent to find the value of theta. In the given range (0 degrees < theta < 90 degrees), we know that tangent is positive, so theta must be in the first quadrant.
We can set up the equation: tan(theta) = opposite/adjacent = 1/2. Since the opposite side is 1 and the adjacent side is 2, we can use the Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 1^2 + 2^2
hypotenuse^2 = 1 + 4
hypotenuse^2 = 5
Taking the square root of both sides, we get:
hypotenuse = √5
Now, we can find cos(theta) using the definition of cosine:
cos(theta) = adjacent/hypotenuse = 2/√5
Therefore, the exact value of cos(theta) is 2/√5 or (√5)/5.
2. If cos(theta) = 8/17, we can find the value of theta by using the inverse cosine function. In the given range (0 degrees < theta < 90 degrees), we know that cosine is positive, so theta must be in the first quadrant.
Using the inverse cosine function, we have:
theta = cos^(-1)(8/17)
To find csc(theta), we can use the reciprocal identity:
csc(theta) = 1/sin(theta)
Since sin(theta) is the opposite/hypotenuse, we need to find the opposite side. We can use the Pythagorean theorem:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = (8)^2 + (17)^2
hypotenuse^2 = 64 + 289
hypotenuse^2 = 353
Taking the square root of both sides, we get:
hypotenuse = √353
Now, we can find sin(theta) using the definition of sine:
sin(theta) = opposite/hypotenuse = 8/√353
Finally, we can find csc(theta) using the reciprocal identity:
csc(theta) = 1/sin(theta) = 1/(8/√353) = √353/8
Therefore, the exact value of csc(theta) is √353/8.
3. If tan(theta) = 1, we can use the definition of tangent to find the value of theta. In the given range (180 degrees < theta < 270 degrees), we know that tangent is negative, so theta must be in the third quadrant.
We can set up the equation: tan(theta) = opposite/adjacent = 1. Since the opposite side is 1 and the adjacent side is -1, we can use the Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 1^2 + (-1)^2
hypotenuse^2 = 1 + 1
hypotenuse^2 = 2
Taking the square root of both sides, we get:
hypotenuse = √2
Now, we can find cos(theta) using the definition of cosine:
cos(theta) = adjacent/hypotenuse = -1/√2
Therefore, the exact value of cos(theta) is -1/√2.
To find the exact value of each expression, we can use the trigonometric ratios and the given information.
1. If tan(theta) = 1/2, we know that the tangent ratio is equal to the ratio of the sine to the cosine. Therefore, we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to find the missing cosine value. In this case, since tangent (theta) = 1/2, we can set up the equation (sin(theta))/cos(theta) = 1/2.
To solve for cos(theta), we first simplify the equation by cross-multiplying: 2*sin(theta) = cos(theta). Now, we can square both sides of the equation: (2*sin(theta))^2 = (cos(theta))^2. Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can rewrite the squared cosine term: (2*sin(theta))^2 = 1 - sin^2(theta).
Expanding the squared sine term and rearranging the equation, we have:
4*sin^2(theta) = 1 - sin^2(theta)
5*sin^2(theta) = 1
sin^2(theta) = 1/5
Taking the square root of both sides, we find: sin(theta) = √(1/5)
Since 0 degrees < theta < 90 degrees, sin(theta) > 0. Therefore, sin(theta) = √(1/5).
Now, we can use the Pythagorean identity to find cos(theta):
cos^2(theta) = 1 - sin^2(theta)
cos^2(theta) = 1 - (1/5)
cos^2(theta) = 4/5
Taking the square root of both sides, we find: cos(theta) = √(4/5). However, since 0 degrees < theta < 90 degrees and cos(theta) > 0 in that quadrant, cos(theta) = √(4/5) = 2/√5 = (2√5)/5.
2. If cos(theta) = 8/17, we can use the Pythagorean identity to find the sine:
sin^2(theta) + cos^2(theta) = 1
sin^2(theta) = 1 - cos^2(theta)
sin^2(theta) = 1 - (8/17)^2
sin^2(theta) = 1 - 64/289
sin^2(theta) = (289 - 64)/289
sin^2(theta) = 225/289
Taking the square root of both sides, we find: sin(theta) = √(225/289). However, since 0 degrees < theta < 90 degrees and sin(theta) > 0 in that quadrant, sin(theta) = √(225/289) = 15/17.
Now, we can find csc(theta) using the reciprocal of the sine:
csc(theta) = 1/sin(theta) = 1/(15/17) = 17/15.
3. If tan(theta) = 1 and 180 degrees < theta < 270 degrees, we can again use the Pythagorean identity to find the cosine:
tan(theta) = sin(theta)/cos(theta)
Since tan(theta) = 1, sin(theta) = cos(theta).
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can substitute sin(theta) for cos(theta):
(sin(theta))^2 + (sin(theta))^2 = 1
2*sin^2(theta) = 1
sin^2(theta) = 1/2
Taking the square root of both sides, we find: sin(theta) = √(1/2).
However, since 180 degrees < theta < 270 degrees and sin(theta) < 0 in that quadrant, sin(theta) = -√(1/2) = -√2/2.
Now, we can use the Pythagorean identity to find cos(theta):
cos^2(theta) = 1 - sin^2(theta)
cos^2(theta) = 1 - (1/2)
cos^2(theta) = 1/2
Taking the square root of both sides, we find: cos(theta) = √(1/2).
However, since 180 degrees < theta < 270 degrees and cos(theta) < 0 in that quadrant, cos(theta) = -√(1/2) = -√2/2.