Question

(i) If cos(40$°$ + x) = sin 30$°$ find the value of x.
(ii) If tan y = sin 45$°$ + cos 45$°$ then find the value of y.

Answer 1

(i)

$$\begin{gathered} \mathrm{Given}:\cos \left(40°+x\right)=\sin 30° \\ \mathrm{We}\mathrm{know}\mathrm{that}\cos \mathit{}x°=\sin (90°-x) \\ \mathrm{Thus},\cos \left(40°+x\right)=\sin \left(90°-40°-x\right) \\ \Rightarrow \cos \left(40°+x\right)=\sin \left(50°-x\right) \\ \Rightarrow \sin 30°=\sin \left(50°-x\right)\left(\mathrm{Given}\right) \\ \Rightarrow 30°=\left(50°-x\right) \\ \Rightarrow x=20° \end{gathered}$$

(ii)

$$\begin{gathered} \mathrm{Given}:\tan y=\sin 45°\cos 45°+\sin 30° \\ \Rightarrow \tan y=\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}+\frac{1}{2} \\ \Rightarrow \tan y=\frac{1}{2}+\frac{1}{2}=1 \\ \Rightarrow \tan y=\tan 45° \\ \Rightarrow y=45° \end{gathered}$$