WHICH EXPRESSION IS EQUIVALENT TO SIN 22 COS 18 + COS 22 SIN 18?
The correct answer and explanation is:
Correct Answer:
sin(22° + 18°) = sin 40°
Explanation:
The expression sin(22°)cos(18°) + cos(22°)sin(18°) is a standard identity in trigonometry. It matches the format of the sine angle addition identity, which is used to simplify expressions involving the sum of two angles.
The sine angle addition identity is:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
In this case:
- Let A = 22°
- Let B = 18°
Substitute these values into the identity:
sin(22° + 18°) = sin(22°)cos(18°) + cos(22°)sin(18°)
This means that:
sin(22°)cos(18°) + cos(22°)sin(18°) = sin(40°)
So, the given expression simplifies directly to sin(40°).
This identity helps in reducing trigonometric expressions involving sums of products to a single sine function, which is easier to evaluate or use in further calculations. Trigonometric identities like this one are useful in calculus, physics, and engineering where angles are often added or subtracted in formulas.
In solving problems, recognizing these patterns saves time and improves accuracy. Rather than evaluating each sine and cosine value separately and then adding them, the identity allows quick simplification to a single trigonometric function value. For example, knowing that sin(40°) is approximately 0.6428, this can replace a longer calculation involving individual sine and cosine evaluations of 22° and 18°.
In summary, the given expression is equivalent to sin(40°) based on the sine angle addition identity.