To determine the exact value of sin(60°) tan(60°) + tan(30°) cos(45°), we will need to use the special triangles:
30°-60°-90° and 45°-45°-90°.
Recall that the ratios of the sides in a 30°-60°-90° triangle are:
- Opposite the 30° angle: x
- Opposite the 60° angle: x√3
- Hypotenuse: 2x
Also, the ratios of the sides in a 45°-45°-90° triangle are:
- Opposite the 45° angles: x
- Hypotenuse: x√2
Using these triangles, we can find the values of the trigonometric functions involved:
sin(60°) = √3/2 (this is the opposite side of the 60° angle over the hypotenuse in the 30°-60°-90° triangle)
tan(60°) = √3 (this is the opposite side of the 60° angle over the adjacent side in the 30°-60°-90° triangle)
tan(30°) = 1/√3 (this is the opposite side of the 30° angle over the adjacent side in the 30°-60°-90° triangle)
cos(45°) = sin(45°) = 1/√2 (this is the adjacent side of the 45° angle over the hypotenuse in the 45°-45°-90° triangle)
Now we can substitute these values into the expression:
sin(60°) tan(60°) + tan(30°) cos(45°) = (√3/2)√3 + (1/√3)(1/√2)
Simplifying this expression, we get:
sin(60°) tan(60°) + tan(30°) cos(45°) = 3/2 + 1/2√2
Therefore, the exact value of sin(60°) tan(60°) + tan(30°) cos(45°) is 3/2 + 1/2√2.