Basic trigonometric equations questions with answers

Questions

1. Question 1Special values

   Solve $\sin x = \dfrac{\sqrt{2}}{2}$ for $0^\circ \leq x \leq 360^\circ$.

   Show the worked solution

   1. Step 1: Basic angle: use the positive value$\alpha = \sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$$= 45^\circ$
   2. Step 2: ASTC (All, Sine, Tan, Cos): in which quadrants does sin have this sign?$\sin x$ is positive in quadrants 1 and 2.
   3. Step 3: List every angle x in the range, quadrant by quadrant$x = 45^\circ$$x = 180^\circ - 45^\circ$$= 135^\circ$
2. Question 2To 1 decimal place

   Solve $\sin x = 0.15$ for $0^\circ \leq x \leq 360^\circ$. Give your answers correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Basic angle: use the positive value$\alpha = \sin^{-1}\left(0.15\right)$$= 8.6^\circ$
   2. Step 2: ASTC (All, Sine, Tan, Cos): in which quadrants does sin have this sign?$\sin x$ is positive in quadrants 1 and 2.
   3. Step 3: List every angle x in the range, quadrant by quadrant$x = 8.6^\circ$$x = 180^\circ - 8.6^\circ$$= 171.4^\circ$
3. Question 3Rearrange first

   Solve $\tan x + 1 = 0$ for $0^\circ \leq x \leq 360^\circ$.

   Show the worked solution

   1. Step 1: Rearrange to ratio = number$\tan x = -1$$\tan x = -1$
   2. Step 2: Basic angle: use the positive value$\alpha = \tan^{-1}\left(1\right)$$= 45^\circ$
   3. Step 3: ASTC (All, Sine, Tan, Cos): in which quadrants does tan have this sign?$\tan x$ is negative in quadrants 2 and 4.
   4. Step 4: List every angle x in the range, quadrant by quadrant$x = 180^\circ - 45^\circ$$= 135^\circ$$x = 360^\circ - 45^\circ$$= 315^\circ$
4. Question 4In radians: exact, in terms of π

   Solve $\cos x = \dfrac{\sqrt{2}}{2}$ for $0 \leq x \leq 2\pi$, giving your answers in terms of $\pi$.

   Show the worked solution

   1. Step 1: Basic angle: use the positive value$\alpha = \cos^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$$= \dfrac{1\pi}{4}$
   2. Step 2: ASTC (All, Sine, Tan, Cos): in which quadrants does cos have this sign?$\cos x$ is positive in quadrants 1 and 4.
   3. Step 3: List every angle x in the range, quadrant by quadrant$x = \dfrac{\pi}{4}$$x = 2\pi - \dfrac{\pi}{4}$$= \dfrac{7\pi}{4}$
5. Question 5In radians: to 2 decimal places

   Solve $\cos x = -0.6$ for $0 \leq x \leq 2\pi$, giving your answers correct to 2 decimal places.

   Show the worked solution

   1. Step 1: Basic angle: use the positive value$\alpha = \cos^{-1}\left(0.6\right)$$= 0.9273$
   2. Step 2: ASTC (All, Sine, Tan, Cos): in which quadrants does cos have this sign?$\cos x$ is negative in quadrants 2 and 3.
   3. Step 3: List every angle x in the range, quadrant by quadrant$x = \pi - 0.9273$$= 2.21$$x = \pi + 0.9273$$= 4.07$