To solve these trigonometric problems, we'll calculate the values step by step. Make sure your calculator is set to degrees (not radians) before you begin.
1.1.1 Calculate cos x + cos y
For x = 36 degrees and y = 56 degrees:
cos x = cos(36°)
cos y = cos(56°)
Now, we add the two values together:
cos x + cos y = cos(36°) + cos(56°)
Using a calculator, you can find:
cos(36°) ≈ 0.809017 (rounded to six decimal places for intermediate calculations)
cos(56°) ≈ 0.559193
Adding these two values gives us:
cos x + cos y ≈ 0.809017 + 0.559193 = 1.368210
Rounded to two decimal places, this result is:
cos x + cos y ≈ 1.37
1.1.2 Calculate 2 tan² (2x + 3y)
First, calculate the angle:
2x + 3y = 2(36°) + 3(56°)
= 72° + 168°
= 240°
Now, find the tangent of this angle:
tan(240°)
Tan(240°) can be calculated using a calculator, the result is negative because tan is positive in the first and third quadrants (240° is in the third quadrant).
Using a calculator:
tan(240°) ≈ -1.732051
Now we'll square this value and multiply by 2:
2 tan²(240°) = 2 (-1.732051)²
= 2 (3.000002)
≈ 6.000004 (rounded for intermediate calculations)
Rounded to two decimal places, this result is:
2 tan² (2x + 3y) ≈ 6.00
1.1.3 Calculate 4 sin (y - x)
First, calculate the angle:
y - x = 56° - 36°
= 20°
Now, find the sine of this angle:
sin(20°)
Using a calculator:
sin(20°) ≈ 0.342020
Multiply this by 4 to get:
4 sin(20°) = 4 0.342020
≈ 1.368080
Rounded to two decimal places, this result is:
4 sin(y - x) ≈ 1.37
In summary, the results are:
1.1.1: cos x + cos y ≈ 1.37
1.1.2: 2 tan² (2x + 3y) ≈ 6.00
1.1.3: 4 * sin(y - x) ≈ 1.37
Remember to use a scientific calculator for these calculations and round off to the nearest two decimal places, as requested.
