8cos30° × tan60°+2 sin²65°+2cos²65°
To simplify this expression, we can use trigonometric identities and basic arithmetic operations.
1. Rewrite the terms using trigonometric identities:
a. 8cos30° can be rewritten as 8 * cos(30°) = 8 * (√3/2) = 4√3.
b. tan60° can be written as sin60°/cos60° = (√3)/(1/2) = 2√3.
c. 2sin²65° can be rewritten as 2 * (sin65°)².
d. 2cos²65° can be rewritten as 2 * (cos65°)².
2. Calculate the values of sin65° and cos65°:
a. sin65° ≈ 0.9063 (rounded to four decimal places).
b. cos65° ≈ 0.4226 (rounded to four decimal places).
3. Substitute the calculated values into the expression:
a. 8cos30° × tan60° + 2sin²65° + 2cos²65° becomes:
4√3 * 2√3 + 2 * (0.9063)² + 2 * (0.4226)².
4. Simplify the expression further:
a. 4√3 * 2√3 = 24 + 2 * (0.9063)² + 2 * (0.4226)².
b. 2 * (0.9063)² = 2 * 0.8215 ≈ 1.643 (rounded to three decimal places).
c. 2 * (0.4226)² = 2 * 0.1790 ≈ 0.358 (rounded to three decimal places).
5. Substitute the simplified values back into the expression:
a. 24 + 1.643 + 0.358 = 25.001.
Therefore, the simplified expression is approximately 25.001.