Introduction to Trigonometry — Exercise 8.2

Substitute exact 30°,45°,60° ratios before simplifying. Work in degrees.

(i)

\[\frac{\sqrt3}{2}\frac{\sqrt3}{2}+\frac12\frac12=1\]

(ii)

\[2(1)^2+\left(\frac{\sqrt3}{2}\right)^2-\left(\frac{\sqrt3}{2}\right)^2=2\]

(iii)

\[\frac{\sqrt2/2}{2/\sqrt3+2}=\frac{\sqrt6}{4(1+\sqrt3)}=\frac{\sqrt6(\sqrt3-1)}8\]

(iv)

\[\frac{1/2+1-2/\sqrt3}{2/\sqrt3+1/2+1}=\frac{3\sqrt3-4}{3\sqrt3+4}\]

(v)

\[\frac{5(1/2)^2+4(2/\sqrt3)^2-1}{(1/2)^2+(\sqrt3/2)^2}=\frac54+\frac{16}3-1=\frac{67}{12}\]

(i) [2/√3]/[1+1/3]=√3/2=sin60°, option A. (ii) (1−1)/(1+1)=0, option D. (iii) At A=0°, sin(2A)=0=2sinA. The other offered angles fail by substitution; option A. (iv) [2/√3]/[1−1/3]=√3=tan60°, option C.

With 0°<A+B≤90° and positive acute A−B, the relations give A+B=60° and A−B=30°. Add: 2A=90°; subtract: 2B=30°.

Use the chapter’s interval 0°≤θ≤90°. (i) False: with A=B=30°, sin60°=√3/2 but sin30°+sin30°=1. (ii) True on this interval: sine increases from 0 to 1. (iii) False: cosine decreases from 1 to 0. (iv) False: at 30°, sine=1/2 and cosine=√3/2; equality holds at 45°, not all angles. (v) True: cot0°=cos0°/sin0°=1/0 is undefined.