# Math Help - Trigonometry Questions and identities.

1. ## Trigonometry Questions and identities.

Could someone please solve these questions. I am struggling with them. Very much appreciated. Donation will be provided

(i) If Cosec theta = 3.2 find theta (in degrees) 0<theta<360

(ii) If sin theta= .7 write down values for
a - sin (180 - theta)
b - sin (360 - theta)

(iii) Using the identity sin^2theta + cos^2theta=1 and the identities for cos(A+B) find:
a - cos 75degrees if cos 45degrees = .707
b - cos 30degrees = 0.866

(iv) Prove that sinx/1-cosx + sinx/1+cosx= 2 cosecx

2. (i) $\csc (\theta) = 3.2$

$\frac{1}{\sin (\theta)}= 3.2$

$1= 3.2 \cdot \sin (\theta)$

$\frac{1}{3.2}=\sin (\theta)$

$\sin^{-1} \bigg( \frac{1}{3.2} \bigg)=\theta$

(ii) $\sin \theta = 0.7$

$\theta = \sin^{-1} (0.7)$

Solve for theta and then you can put it into your other 2 equations. What are the values of a and b?

3. (iv) $\frac{\sin x}{1-\cos x} + \frac{\sin x}{1 + \cos x}$

Cross multiply to get the same denominator:

$=\dfrac{(\sin x)(1+ \cos x)}{(1+ \cos x)(1-\cos x)} + \dfrac{(\sin x)(1-\cos x)}{(1 + \cos x)(1 - \cos x)}$

$=\dfrac{(\sin x)(1+ \cos x)+(\sin x)(1-\cos x)}{(1+ \cos x)(1-\cos x)}$

Expand brackets and group like terms:

$=\dfrac{2 \sin (x)}{1 - \cos^2 (x)}$

Now use the pythagorean identity $\sin^2 x + cos^2 x = 1$ to get your answer.

4. Thankyou so much for your help. I was able to understand what i actually had to do much better. I am fairly new to trigonometry.

For the proof would this be correct. following on from the last line you gave me.

2sin(x)/1-cos^2(x)
{sin(x)/1-cos(x)} + {sin(x)/1+cos(x)} = 2sec(x)

5. That doesn't prove the trigonometric identity yet, and it is also incorrect (it should be 2csc(x)).

Remember that $2 \csc x = \frac{2}{\sin x}$

What I had in mind was substituting $1 - \cos^2 x$ with $\sin^2 x$ and the simplifying to get your answer.

6. ok sorry i'm not very confident with trig. So:

2sin(x)/1-cos^2(x)=2csc(x)
2sin(x)/sin^2(x) =2csc(x)
2/sin(x) =2csc(x)

Is this now a proven identity

7. Yes!

Because csc(x)=1/sin(x)