use the given information to find the exact values of the remaining trigonometric functions of theta. csc(theta)=5 with pi/2 less than theta less than pi

Question

i need answers for cos(theta), tan(theta), sec(theta) and cot(theta)

 

Answer 1

form a right triangle in the second quadrant because θ is between 90º and 180º

cscθ=5

 

csc is the reciprocal of the sin

sinθ=1/5

sin=opposite/ hypotenuse

therefore the adjacent side is "a" where a^2+1^2=5^2

a^2+1=25

a^2=25-1

a^2=24

a=√24

a=2√6

cos=adjacent/hypotenuse

cosθ=2√6/5 but cos is negative in the 2nd quadrant

cosθ=-2√6/5

tan=opposite/adjacent

tanθ=1/2√6

tanθ=√6/12

tan is negative in the second quadrant

tanθ=-√6/12

sec=1/cos

secθ=-5/2√6

secθ=-5√6/12

cot=1/tan

cotθ=-12/√6

cotθ=-12√6/6

cotθ=-2√6

Answer 2

The interval pi/2 < theta < pi tells us that theta is in the second quadrant. In the second quadrant, the sine and cosecant functions are positive. The cosine and secant functions are negative. Consequently, the tangent and cotangent functions are also negative in the second quadrant, since the tangent is the sine divided by the cosine and the cotangent is the cosine divided by the sine.

 

We also know that the cosecant is the hypotenuse divided by the opposite side of the right triangle. If we know two sides of the right triangle, we can find the third side by using the Pythagorean Theorem a² + b² = c², where a and b are the legs of the right triangle, and c is the hypotenuse.

 

In our problem, the hypotenuse is 5, and one of the legs is equal to 1. To find the third leg, substitute these values into the Pythagorean Theorem to find the remaining leg. We will let a represent the missing third leg.

 

a² + 1² = 5²

 

a² + 1 = 24

 

a² = 24

 

a = √24 = 2√6

 

Now that you know the lengths of each of the three sides of the right triangle, can you determine the values of the remaining trig functions, Paige?