Introduction to Trigonometry Set-1

Q1. In a right-angled triangle, if $\sin\theta=\dfrac{3}{5}$ and $\theta$ is acute, what is the value of $\tan\theta$?

Q2. If $\cos\theta=\dfrac{12}{13}$ with $\theta$ acute, find $\sin2\theta$.

Q3. Given $\tan\theta=\dfrac{4}{3}$ and $\theta$ is acute, the value of $\dfrac{1+\sin\theta}{\cos\theta}$ equals:

Q4. From two points A and B on the same straight line towards the base of a vertical tower, the angles of elevation to the top are $30^\circ$ at A and $60^\circ$ at B. If $AB=30\ \text{m}$ and B lies between A and the tower, the height of the tower is:

Q5. Solve for $0^\circ<\theta<90^\circ$: $\sin\theta=\cos2\theta$. The value of $\theta$ is:

Q6. In a right triangle, $\tan\theta = \dfrac{3}{4}$. Find $\sin\theta$.

Q7. In triangle $ABC$, $\angle B = 90^\circ$ and the altitude from $B$ meets hypotenuse $AC$ at $D$. If $AD=9$ and $DC=16$, find $\tan A$.

Q8. If $\tan\theta + \cot\theta = \dfrac{5}{2}$ and $\theta$ is acute with $\theta>45^\circ$, find $\sec^2\theta$.

Q9. If $\tan\frac{\theta}{2} = \dfrac{2}{3}$ and $0<\theta<\pi$, find $\sin\theta$.

Q10. For acute angles $A$ and $B$ with $A\neq B$, it is given that $\cos A + \cos B = \sin A + \sin B$. Which of the following must be true?

Q11. In right-angled triangle ABC (right angle at B), the hypotenuse AC = $12$ and $\angle A = 30^\circ$. Find the length of side BC (side opposite $\angle A$).

Q12. From a point $P$ on level ground, the angle of elevation of the top of a tower is $30^\circ$. After walking $30\,$m towards the tower to point $Q$, the angle of elevation becomes $45^\circ$. Find the height of the tower.

Q13. If $\cos\theta=-\dfrac{3}{5}$ and $\theta\in\left(\dfrac{\pi}{2},\pi\right)$, then the value of $\sin\left(\dfrac{\theta}{2}\right)$ is

Q14. Solve the equation $\sec\theta=2+\tan\theta$ for $\theta\in[0,2\pi)$. (Give the principal solution in terms of inverse trig.)

Q15. In triangle ABC, angles satisfy $A=2B$ and $\cot B=3$. The value of $\tan A$ is