Find csc(theta), tan (theta), and cos (theta), where theta is the angle shown in the figure.
Give exact values, not decimal approximations
c=10
b=7
a=7.14
the right angle is locate between sides a and b and the theta angle is an acutle angle sides b and c.
i have sin theta as 500/357
i have csc theta as 7/10
i have tan theta as 51/50
sin T = 714/1000
so
csc T =1/sin T= 1000/714 = 500/357
tan T = 714/700 = 102/100= 51/50
cos T = 7/10
You have them right but mislabeled.
To find the values of csc(theta), tan(theta), and cos(theta), we can use the given side lengths of the triangle.
First, let's identify the sides of the triangle based on the information given:
- The side opposite angle theta is side b, which has a length of 7.
- The side adjacent to angle theta is side a, which has a length of 7.14.
- The hypotenuse of the triangle is side c, which has a length of 10.
Now, we can find the values of the trigonometric functions:
1. csc(theta) (cosecant):
Recall that cosecant is the reciprocal of the sine function. We can find the sine of angle theta by using the ratio of the side opposite to theta (b) and the hypotenuse (c):
sin(theta) = b/c = 7/10
Taking the reciprocal, we get:
csc(theta) = 1/sin(theta) = 1/(7/10) = 10/7
Therefore, csc(theta) = 10/7.
2. tan(theta) (tangent):
Tangent is calculated by dividing the side opposite angle theta (b) by the side adjacent to it (a):
tan(theta) = b/a = 7/7.14
To simplify the fraction, we can multiply both the numerator and the denominator by 100 to get rid of the decimal:
tan(theta) = (7/7.14) * (100/100) = 700/714
To simplify further, we can divide both numerator and denominator by their greatest common divisor (which is 14):
tan(theta) = (700/14) / (714/14) = 50/51
Therefore, tan(theta) = 50/51.
3. cos(theta) (cosine):
To find the cosine value, we can use the Pythagorean theorem. Since we know the lengths of sides a and c, we can find the remaining side (side opposite the right angle) using the equation:
a^2 + b^2 = c^2
(7.14)^2 + 7^2 = 100 + 49 = 149
Now we can find cos(theta) by dividing the side adjacent to the angle (a) by the hypotenuse (c):
cos(theta) = a/c = 7.14/10
To simplify the fraction, we can multiply both the numerator and denominator by 100 to get rid of the decimal:
cos(theta) = (7.14/10) * (100/100) = 714/1000
To further simplify, we can divide both numerator and denominator by their greatest common divisor (which is 2):
cos(theta) = (714/2) / (1000/2) = 357/500
Therefore, cos(theta) = 357/500.
In summary, the exact values of the trigonometric functions are:
csc(theta) = 10/7
tan(theta) = 50/51
cos(theta) = 357/500.
To find the values of csc(theta), tan(theta), and cos(theta), we can use the given information.
We can start by using the Pythagorean theorem to find side c:
c² = a² + b²
c² = (7.14)² + (7)²
c² = 51.1396 + 49
c² = 100.1396
√c² = √100.1396
c = 10 (rounded to two decimal places)
To find sin(theta), we can use the given ratio: sin(theta) = opposite/hypotenuse
sin(theta) = b/c
sin(theta) = 7/10
Now we can find csc(theta) by taking the reciprocal of sin(theta): csc(theta) = 1/sin(theta)
csc(theta) = 1 / (7/10)
csc(theta) = 10/7
To find tan(theta), we can use the given ratio: tan(theta) = opposite/adjacent
tan(theta) = b/a
tan(theta) = 7/7.14
Finally, we can find cos(theta) using the Pythagorean theorem: cos(theta) = adjacent/hypotenuse
cos(theta) = a/c
cos(theta) = 7.14/10