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
∴ \(\cos {{22.5}^{\circ }}=\sqrt{\frac{1+\cos {{45}^{\circ }}}{2}}\).
\(=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}}\).
\(=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}}\).
\(=\frac{\sqrt{2+\sqrt{2}}}{2}\).
\(\sin {{22.5}^{\circ }}=\sqrt{\frac{1-\cos {{45}^{\circ }}}{2}}=\frac{\sqrt{2-\sqrt{2}}}{2}\).
\(\tan {{22.5}^{\circ }}=\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}=\frac{2-\sqrt{2}}{\sqrt{2}}=\sqrt{2}-1\).
\(\cot {{22.5}^{\circ }}=\frac{\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}=\sqrt{2}+1\).