Evaluate cos(pi/3)^2
The problem asks for the computation of the square of the cosine of an angle whose measure is pi/3 radians. Specifically, you need to find the value of cos(pi/3) first and then square that result to get the final answer.
$\left(cos\right)^{2} \left(\right. \frac{\pi}{3} \left.\right)$
Answer
Solution:
Step 1:
Determine the value of $\cos\left(\frac{\pi}{3}\right)$, which is $\frac{1}{2}$. Then square this value: $\left(\frac{1}{2}\right)^2$.
Step 2:
Square the numerator and the denominator separately: $\frac{1^2}{2^2}$.
Step 3:
Any number raised to the power of zero is one, so $1^2$ remains $1$: $\frac{1}{2^2}$.
Step 4:
Calculate the square of $2$, which is $4$: $\frac{1}{4}$.
Step 5:
Express the final result in both exact and decimal forms: Exact Form: $\frac{1}{4}$, Decimal Form: $0.25$.
Knowledge Notes:
Trigonometric Functions: The cosine function relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. For specific angles, such as $\frac{\pi}{3}$, the cosine values are known exactly.
Squaring a Fraction: To square a fraction, square both the numerator and the denominator. In this case, $(\frac{a}{b})^2 = \frac{a^2}{b^2}$.
Exponent Rules: When raising a number to the power of 2, you multiply the number by itself. For example, $2^2 = 2 \times 2$.
Exact vs Decimal Form: An exact form of a number is a representation that captures the number's value without any approximation, often as a fraction. A decimal form is a representation of the number in base 10, which may be an approximation if the number is irrational or if the decimal is truncated or rounded.
