By means of fundamental identities, express each of the following in terms of sinθ only: tanθShow complete solution and explain the answer.
You need to remember what basic trigonometric identity is:
`sin^2theta + cos^2theta = 1`
You need to remember that the tangent function is rational such that: `tan theta` = `sin theta/cos theta`
Hence, you need to form tangent function in the basic trigonometric identity by dividing identity by `cos^2 theta` . You need to divide both sides by `cos^2 theta` to preserve the identity.
`(sin^2 theta)/(cos^2 theta) + 1 = 1/(cos^2 theta)`
`tan^2 theta + 1 = 1/(cos^2 theta)`
You need to write cos alpha in terms of sin alpha such that:
`cos^2 theta= 1 - sin^2 theta`
`tan^2 theta = 1/(1 - sin^2 theta) - 1`
Bringing the terms to a common denominator yields:
`tan^2 theta = (1 - 1 + sin^2 theta)/(1 - sin^2 theta)`
`tan^2 theta = (sin^2 theta)/(1 - sin^2 theta)`
`tan theta = +- (sin theta)/(sqrt(1 - sin^2 theta))`
Hence,writing the tangent function in terms of sin theta yields:
`tan theta = +- (sin theta)/(sqrt(1 - sin^2 theta)).`