Rewrite the expression tan(90°−θ)sinθ as one of the six trigonometric functions of acute angle thetaθ.
Please show work + answer. I have a few of them to do. Thanks.
tan(90º - Θ) = cotan(Θ)
cotan(Θ) = cos(Θ) / sin(Θ)
[cos(Θ) / sin(Θ)] * sin(Θ) = cos(Θ)
To rewrite the expression `tan(90°−θ)sinθ` as one of the six trigonometric functions, we can use trigonometric identities.
First, let's rewrite `tan(90°−θ)` using the identity `tan(90°−θ) = cotθ`:
`tan(90°−θ)sinθ = cotθsinθ`
Next, let's simplify `cotθsinθ` using the identity `cotθ = cosθ/sinθ`:
`cotθsinθ = (cosθ/sinθ)sinθ`
The `sinθ` terms cancel out:
`= cosθ`
Therefore, `tan(90°−θ)sinθ` is equivalent to `cosθ`.
To rewrite the expression tan(90° - θ)sinθ as one of the six trigonometric functions, we can use trigonometric identities.
First, let's recall two important trigonometric identities:
1. The cofunction identity: sin(90° - θ) = cosθ
2. The quotient identity: tanθ = sinθ / cosθ
Using these identities, we can rewrite the expression:
tan(90° - θ)sinθ
= (sin(90° - θ) / cos(90° - θ)) * sinθ
Now, substitute the cofunction identity sin(90° - θ) = cosθ:
= (cosθ / cos(90° - θ)) * sinθ
Next, recall another trigonometric identity: sinθ = 1/cscθ:
= (cosθ / cos(90° - θ)) * (1/cscθ)
Applying the cofunction identity cos(90° - θ) = sinθ:
= (cosθ / sinθ) * (1/cscθ)
Using the reciprocal identity cscθ = 1/sinθ:
= cosθ * (1/sinθ) * (1/cscθ)
This simplifies to:
= cosθ / sinθ
Finally, using the reciprocal identity for sine, sinθ = 1/cscθ:
= cosθ * cscθ
Therefore, the expression tan(90° - θ)sinθ is equivalent to cosθ * cscθ.