# Thread: trigonometric functions

1. ## trigonometric functions

Math trigonometric questions?

1)Determine tan2θ if sinθ + cosθ = -1

2) Express sin2θ in terms of csc θ and sec θ

3) Express cos3θ in terms of sin θ

2. Originally Posted by crosser43
Math trigonometric questions?

1)Determine tan2θ if sinθ + cosθ = -1

2) Express sin2θ in terms of csc θ and sec θ

3) Express cos3θ in terms of sin θ
1) sinθ + cosθ = -1
sinθ + cosθ + 1 = 0
2sin(0.5θ)cos(0.5θ) + 2cos^2(0.5θ) = 0
2*cos(0.5θ)[sin(0.5θ) + cos(0.5θ)] = 0
cos(0.5θ) = 0 or 0.5θ = 0.5π or θ = π
sin(0.5θ) + cos(0.5θ) = 0
tan(0.5θ) = -1 or 0.5θ = 0.25π or 2θ = π

3. ## Trig

Hello crosser43
Originally Posted by crosser43
Math trigonometric questions?

1)Determine tan2θ if sinθ + cosθ = -1
Square both sides:

$(\sin\theta+\cos\theta)^2=1$

$\Rightarrow \sin^2\theta +2\sin\theta\cos\theta+\cos^2\theta = 1$

Now use the identities $\sin^2\theta +\cos^2\theta = 1$ and $2\sin\theta\cos\theta = \sin2\theta$:

$\Rightarrow 1 + \sin2\theta = 1$

$\Rightarrow \sin2\theta = 0$

$\Rightarrow \tan2\theta = \frac{\sin2\theta}{\cos2\theta}=0$

2) Express sin2θ in terms of csc θ and sec θ
$\sin2\theta= 2\sin\theta\cos\theta = 2\left(\frac{1}{\csc\theta}\right)\left(\frac{1}{\ sec\theta}\right)=\frac{2}{\csc\theta\sec\theta}$

3) Express cos3θ in terms of sin θ
See line 40 of Multiple-Angle Formulas -- from Wolfram MathWorld, and put $\cos\theta = \sqrt{1-\sin^2\theta}$