Let t be an angle between π and 3π/2, and suppose sec(t)=−2.6. What is tan(t)?

Question

Let t be an angle between π and 3π/2, and suppose sec(t)=−2.6. What is tan(t)?

Having a hard time with this one..

Answer 1

sec(t) = -2.6 = -26/10 = -13/5

so cos(t) = -5/13
construct your right-angled triangle with hypotenuse 13 and adjacent side of 5
( remember cosØ = adjacent/hypotenuse )
so 5^2 + y^2 = 13^2
y^2 = 169-25 = 144
y = ± 12
but for π < t < 3π/2 , y = -12
Tan(t) = opposite/ adjacent = -12/5

Answer 2

thank you so much!!! Also, I have one more question..

Assume r>0. Which of the following equals csc(tan−1(r))?
Is sqrt1+r2 the correct answer

Answer 3

angle t is in quadrant II, so the x is negative, and the y is positive,

so x = -5, and y = 12

does not change the final answer of tan(t) = -12/5

Answer 4

csc(tan−1(r)) = ???

start with tan^-1 (r) = tan^-1 (r/1)
so you have a triangle with opposite of r and adjacent of 1
then hypotenuse^2 = r^2 + 1, hyp = √(r^2+1)
so sin(tan^−1 (r)) = opposite/hypo = r/√(r^2+1)

and csc(tan^−1 (r)) = √(r^2+1) / r

Answer 5

Quad III ... tan is positive

identity ... tan^2 = sec^2 - 1

Answer 6

scott is correct, the angle was in III , (time to go to bed)

make the necessary changes, or follow scott's relation

tan^2 = sec^2 - 1

= (-13/5)^2 - 1 = 144/25
= + 12/5 in quad III

Answer 7

To find the value of tan(t) when sec(t) = -2.6, we can use the relationship between sec(x) and tan(x).

First, let's recall the definitions of secant (sec) and tangent (tan) of an angle:

sec(t) = 1 / cos(t)
tan(t) = sin(t) / cos(t)

Given sec(t) = -2.6, we can substitute it into the equation of sec(t):

1 / cos(t) = -2.6

To solve for cos(t), we need to isolate it by taking the reciprocal of both sides of the equation:

cos(t) = 1 / (-2.6)
cos(t) = -0.3846

Now that we have the value of cos(t), we can use it to find sin(t):

To get sin(t), we can use the Pythagorean identity:
sin^2(t) + cos^2(t) = 1

Substituting the value of cos(t) we found earlier:

sin^2(t) + (-0.3846)^2 = 1
sin^2(t) + 0.1479 = 1
sin^2(t) = 1 - 0.1479
sin^2(t) = 0.8521

Taking the square root of both sides, we find the value of sin(t):

sin(t) = ± √0.8521
sin(t) = ± 0.9228

Now that we know the values of both sin(t) and cos(t), we can find the value of tan(t) by dividing sin(t) by cos(t):

tan(t) = sin(t) / cos(t)
tan(t) = (± 0.9228) / (-0.3846)

Therefore, the value of tan(t) when sec(t) = -2.6 is approximately ±2.396.