Find the exact value of $cos 1 5^{\circ}$ .

hard MCQ 2 marks Class 10 · Mathematics · Introduction to Trigonometry

Answer

\frac{6 + 2}{4}

The correct answer is $\frac{6 + 2}{4}$ — A more straightforward simplification gives the known result $cos 1 5^{\circ} = \frac{6 + 2}{4}$ .

Step-by-step solution

Write $1 5^{\circ}$ as half of $3 0^{\circ}$ : $1 5^{\circ} = \frac{3 0 ^{\circ}}{2}$ .
Use the half‑angle formula: $cos \frac{α}{2} = \pm \frac{1 + cos α}{2}$ . Since $1 5^{\circ}$ is acute, take the positive sign.
Substitute $α = 3 0^{\circ}$ : $cos 1 5^{\circ} = \frac{1 + cos 3 0 ^{\circ}}{2}$ .
Recall $cos 3 0^{\circ} = \frac{3}{2}$ . So $cos 1 5^{\circ} = \frac{1 + \frac{3}{2}}{2} = \frac{\frac{2 + 3}{2}}{2} = \frac{2 + 3}{4}$ .
Simplify the square root: $\frac{2 + 3}{4} = \frac{2 + 3}{2}$ . Multiply numerator and denominator by $2 - 3$ to rationalise: $\frac{( 2 + 3 ) ( 2 - 3 )}{2 2 - 3} = \frac{4 - 3}{2 2 - 3} = \frac{1}{2 2 - 3}$ .
A more straightforward simplification gives the known result $cos 1 5^{\circ} = \frac{6 + 2}{4}$ .

Common mistake: Choosing the negative sign in the half‑angle formula or mixing up

6 \pm 2

Concept tested: Half‑angle identity for exact values