Use the Pythagorean identities rather than reference triangles . Find tanθ and cotθ if secθ = 5/2 and sinθ <0.
Please help ! Much appreciated !!
Given any one trigonometric relation, you can obtain the other five using simple identities.
If secθ = 5/2,
(secθ)^2 = (tanθ)^2 + 1 (Trigonometric Identity)
=> (5/2)^2 = (tanθ)^2 + 1
=> 25/4 - 1 = (tanθ)^2
=> 21/4 = (tanθ)^2
If sinθ is negative, and cosθ is positive,
=> tanθ = -(root(21/4))
Now, cotθ = (1/tanθ)
= (1/(21/4))
= -(root(4/21))
To find tanθ and cotθ, we can start by using the given information that secθ = 5/2.
The Pythagorean identity for secant is:
sec²θ = 1 + tan²θ
We can substitute the given value of secθ = 5/2 into the equation:
(5/2)² = 1 + tan²θ
25/4 = 1 + tan²θ
Next, we can rearrange the equation to solve for tan²θ:
tan²θ = 25/4 - 1
tan²θ = 25/4 - 4/4
tan²θ = 21/4
Taking the square root of both sides, we get:
tanθ = ±√(21/4)
Since sinθ is given to be less than 0, we know that tangent is negative in the corresponding quadrant (either in the second or fourth quadrant).
Therefore, we have:
tanθ = -√(21/4) or tanθ = √(21/4)
To find cotθ, we can use the identity:
cotθ = 1/tanθ
Substituting the values we found for tanθ, we have:
cotθ = 1/(-√(21/4)) or cotθ = 1/√(21/4)
To simplify, we multiply the numerator and denominator by 2:
cotθ = -2/√21 or cotθ = 2/√21
Therefore, tanθ can be either -√(21/4) or √(21/4), and cotθ can be either -2/√21 or 2/√21.