Solution: Step 1: Determine the value of cosleft(fracpi3right), which is frac12. Then square this value: left(frac12right)2. Step 2: Square the numerator and the denominator separately: frac1222. Step 3: Any number raised to the power of zero is one, so 12 remains 1: frac122. Step 4: Calculate the square of 2, which is 4: frac14. Step 5: Express the final result in both exact and decimal forms: Exact Form: frac14, Decimal Form: 0.25. Knowledge Notes: 1. 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 fracpi3, the cosine values are known exactly. 2. Squaring a Fraction: To square a fraction, square both the numerator and the denominator. In this case, (fracab)2 = fraca2b2. 3. Exponent Rules: When raising a number to the power of 2, you multiply the number by itself. For example, 22 = 2 times 2. 4. 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.

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.

${(c o s)}^{2} (\frac{π}{3})$

Answer

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Solution:

Step 1:

Determine the value of $\cos (\frac{π}{3})$ , which is $\frac{1}{2}$ . Then square this value: ${(\frac{1}{2})}^{2}$ .

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 $\frac{π}{3}$ , 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.