Trigonometry (Intermediate)

Which of the following graphs represent the function  $f\left(x\right)=\cos\left(2x\right)$ ?

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Which of the following is true?

• Sine is positive in the first and third quadrants.

• Cosine is negative in the first and fourth quadrants.

• Tangent is negative in the second and fourth quadrants.

• Sine is negative in the second and third quadrants.

Calculate the value(s) of  $\theta$ such that  $\cos\left(\theta\right)=-\frac{1}{2}$ for  $0\le\theta\le2\pi$ .  

•  $\theta=\frac{2\pi}{3},\frac{4\pi}{3}$  

•  $\theta=\frac{\pi}{3},\frac{2\pi}{3}$  

•  $\theta=\frac{2\pi}{3}$  

•  $\theta=\frac{\pi}{3},\frac{5\pi}{3}$  

Calculate the value(s) of  $\theta$ such that  $\tan\left(2\theta\right)=\sqrt{3}$  for  $0\le\theta\le2\pi$ . 

•  $\theta=\frac{\pi}{6},\frac{2\pi}{3}$  

•  $\theta=\frac{\pi}{6},\frac{2\pi}{3},\frac{\ 7\pi}{6},\frac{6\pi}{5}$  

•  $\theta=\frac{\pi}{3},\frac{4\pi}{3},\frac{\ 7\pi}{3},\frac{12\pi}{5}$  

•  $\theta=\frac{\ 7\pi}{6},\frac{6\pi}{5}$  

Solve the equation $2\sin\left(\theta\right)=\tan\left(\theta\right)$ for  $0^{\circ}\le\theta\le360^{\circ}$ . 
Note: Use the fact that $\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$ .

•  $\theta=0^{\circ},30^{\circ},90^{\circ},150^{\circ},180^{\circ}$  

•  $\theta=0^{\circ},180^{\circ},360^{\circ}$  

•  $\theta=0^{\circ},60^{\circ},180^{\circ},300^{\circ},360^{\circ}$  

•  $\theta=0^{\circ},300^{\circ}$  

For $0^{\circ}\le\theta\le360^{\circ}$ , solve  $6\cos^2\left(\theta\right)+6\cos\left(\theta\right)-1=0$ . 

•  $\theta=81.6^{\circ},120^{\circ},240^{\circ},278.4^{\circ}$  

•  $\theta=81.6^{\circ},120^{\circ}$  

•  $\theta=240^{\circ},120^{\circ}$  

•  $\theta=81.6^{\circ},\ 278.4^{\circ}$  

Solve the equation $\sin^2\left(\theta\right)+3\cos\left(2\theta\right)=2$ for  $0\le\theta\le360\degree^{ }$ . Give your answer(s) to 1 decimal place.  
Note: Use the fact that  $\cos\left(2\theta\right)=1-2\sin^2\left(\theta\right)$  

•  $\theta=153.4^{\circ}$  

•  $\theta=26.6^{\circ}$  

•  $\theta=26.6^{\circ},153.4^{\circ}$  

•  $\theta=26.6^{\circ},153.4^{\circ},206.6^{\circ},333.4^{\circ}$  

A right-angled triangle XYZ has an angle, $\theta$ , where  $\sin\left(\theta\right)=\frac{\sqrt{5}}{5}$ . Without evaluating $\theta$ , calculate the exact value(in surd form if applicable) of $\cos\left(\theta\right)$ .    

•  $\cos\left(\theta\right)=2\sqrt{5}$  

•  $\cos\left(\theta\right)=\frac{2}{5}$  

•  $\cos\left(\theta\right)=\frac{\sqrt{5}}{5}$  

•  $\cos\left(\theta\right)=\frac{2\sqrt{5}}{5}$  

A right-angled triangle XYZ has an angle, $\theta$ , where  $\sin\left(\theta\right)=\frac{\sqrt{5}}{5}$ . Without evaluating $\theta$ , calculate the exact value(in surd form if applicable) of $\sin\left(2\theta\right)$ .    
Note: Use the fact that  $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$ . 

•  $\sin\left(2\theta\right)=\frac{4}{5}$  

•  $\sin\left(2\theta\right)=\frac{1}{5}$  

•  $\sin\left(2\theta\right)=\frac{2\sqrt{5}}{5}$  

•  $\sin\left(2\theta\right)=\frac{\sqrt{10}}{5}$  

A wire in the form of a circle with radius 4 cm is reshaped into the form of a sector of a circle with radius 10 cm. Determine, in radians, the angle of the sector, giving your answer in terms of $\pi$ . 

•  $\theta=\frac{4\pi}{5}-1$  

•  $\theta=\frac{2\left(2\pi-5\right)}{5}$  

•  $\theta=\frac{4\pi}{5}$  

•  $\theta=4\left(2\pi-5\right)$