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Use the special triangles to determine exact value of:

sin(60°) tan(60°) + tan(30°) cos(45°)

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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.

User Cerin
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