# Values of Trigonometric Ratios of Typical Angles – I

## Values of Trigonometric Ratios of Typical Angles – I

(i) Value of 7½°

θ = 7½° then 2θ = 15°

$$\tan \theta =\frac{1-\cos 2\theta }{\sin 2\theta }$$ [∵ 1-cos2θ = 2sin²θ and sin2θ = 2sinθcosθ].

$$=\frac{1-\cos {{15}^{\circ }}}{\sin {{15}^{\circ }}}=\frac{1-\frac{\sqrt{3}+1}{2\sqrt{2}}}{\frac{\sqrt{3}-1}{2\sqrt{2}}}$$.

$$=\frac{2\sqrt{2}-\sqrt{3}-1}{\sqrt{3}-1}$$.

= (√3 – √2) (√2 – 1)

(ii) Value of cot 82½

cot 82½° = cos (90° – 7½°)

= tan 7½°

= (√3 – √2) (√2 – 1)

(iii) Value of cot 7½°

Let θ = 7½° then 2θ = 15°

Now, $$\cot \theta =\frac{1+\cos 2\theta }{\sin 2\theta }$$.

$$=\frac{1+\cos {{15}^{\circ }}}{\sin {{15}^{\circ }}}$$.

$$=\frac{1+\frac{\sqrt{3}+1}{2\sqrt{2}}}{\frac{\sqrt{3}-1}{2\sqrt{2}}}$$.

$$=\frac{2\sqrt{2}+\sqrt{3}+1}{\sqrt{3}-1}$$.

= (√3 + √2) (√2 + 1)

(iv) Value of tan 82½°

tan 82½° = tan (90° – 7½°)

= cot 7½°

= (√3 + √2) (√2 + 1)

2. Value of trigonometric functions for θ = 22.5°

We know that cos 2θ = 2cos² θ – 1

∴ θ = 22.5°.

we have cos 45° = 2cos² 22.5°-1

∴ $$\cos {{22.5}^{\circ }}=\sqrt{\frac{1+\cos {{45}^{\circ }}}{2}}$$.

$$=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}}$$.

$$=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}}$$.

$$=\frac{\sqrt{2+\sqrt{2}}}{2}$$.

$$\sin {{22.5}^{\circ }}=\sqrt{\frac{1-\cos {{45}^{\circ }}}{2}}=\frac{\sqrt{2-\sqrt{2}}}{2}$$.

$$\tan {{22.5}^{\circ }}=\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}=\frac{2-\sqrt{2}}{\sqrt{2}}=\sqrt{2}-1$$.

$$\cot {{22.5}^{\circ }}=\frac{\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}=\sqrt{2}+1$$.