Basics of Trigonometry (Level - 2)
Q1β10 of 50Find the exact value of tan(15Β°).
Solution: Express 15Β° as (45Β°β30Β°) and use tan(AβB) formula. tan15Β° = (β3β1)/(β3+1) = 2ββ3
Find the exact value of cos(22.5Β°).
Use the cosine half-angle formula $\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos\theta}{2}}$ with $\theta = 45^\circ$.
$\cos(22.5^\circ) = \sqrt{\frac{1 + \sqrt{2}/2}{2}} = \frac{\sqrt{2 + \sqrt{2}}}{2}$.
Evaluate sin75Β°cos15Β° β cos75Β°sin15Β°.
Evaluate tan(22.5Β°) + cot(22.5Β°).
Solution: tanΞΈ+cotΞΈ = 2/sin2ΞΈ. For ΞΈ=22.5Β°: 2/sin45Β° = 2β2.
Find the exact value of sin(15Β°) + cos(15Β°).
Solution: sin15Β°=(β6ββ2)/4, cos15Β°=(β6+β2)/4. Sum = β6/2.
Evaluate the expression:
Solution: This is tan(AβB) with A=75Β°,B=15Β°: tan60Β° = β3.
Find the exact value of tan(67.5Β°) β tan(22.5Β°).
Solution: tan67.5Β° = β2+1, tan22.5Β° = β2β1. Difference = 2
Evaluate 8sin15Β°.cos15Β°.cos30Β°.
Find the value of:
Solution: This equals cos(2Β·22.5Β°) = cos45Β° = β2/2.
Evaluate cos(22.5Β°).cos(67.5Β°).
Solution: cos67.5Β°=sin22.5Β°, so expression = sin22.5Β°cos22.5Β° = sin45Β°/2 = β2/4.