Question Type 2: Given an equation of a line, finding the angle of elevation Exercises

Find the acute angle of inclination (in degrees) between the line $y = -3x + 4$ and the positive $x$-axis.

Find the angle of elevation $\varphi$ (in degrees) of the line $y = 25x + 9$ above the positive $x$-axis.

Simplify the expression $\frac{\tan\theta}{\sec\theta}$ in terms of $\sin\theta$ and $\cos\theta$.

Prove the identity $\tan^2\theta + 1 = \sec^2\theta$.

Find the angle of inclination $\varphi$ (in degrees) of the line given by $3x - 4y + 12 = 0$ with respect to the positive $x$-axis.

Express $\displaystyle \frac{1}{\sin x\,\tan x}$ in terms of $\cos x$ only.

Simplify the expression $\frac{\tan\theta\,(1 + \cos\theta)}{\sin\theta}$.

Calculate the acute angle (in degrees) between the lines $y = 2x + 1$ and $y = -\frac{1}{3}x - 3$.

Simplify the expression $\displaystyle \frac{1 - \cos^2 x}{\cos x\,\sin x}$.

Solve the equation $3\sin x - 4\cos x = 0$ for $x$ in the interval $[0,2\pi)$.

Rearrange and simplify the equation $\sec x - \tan x = \cos x$ to an identity involving only $\sin x$ and $\cos x$.

Solve the equation $\tan^2 x - 3\tan x + 2 = 0$ for $x$ in $[0,\pi)$.

Simplify the following to an expression involving only $\tan\theta$: $\frac{\tan^2\theta - \sin^2\theta}{\sin^2\theta}\,.$

Prove the identity $\tan x - \sin x = \frac{\sin^2 x}{\cos x\,(1 + \cos x)}.$