What is the single trigonometric function of (Cos 20 degrees) (Cos 30 degrees)-(Sine 20 degress) (Sin 30 degrees)
To find the value of the expression (Cos 20 degrees) (Cos 30 degrees) - (Sine 20 degrees) (Sin 30 degrees), follow these steps:
Step 1: Calculate the values of cos 20 degrees and cos 30 degrees.
- cos 20 degrees ≈ 0.9397
- cos 30 degrees ≈ 0.8660
Step 2: Calculate the values of sin 20 degrees and sin 30 degrees.
- sin 20 degrees ≈ 0.3420
- sin 30 degrees ≈ 0.5000
Step 3: Substitute the values into the expression: (0.9397) (0.8660) - (0.3420) (0.5000)
Step 4: Calculate the expression: (0.9397) (0.8660) - (0.3420) (0.5000) ≈ 0.8116 - 0.1710 ≈ 0.6406
The single trigonometric function of (cos 20 degrees) (cos 30 degrees) - (sin 20 degrees) (sin 30 degrees) is approximately 0.6406.
To find the value of the expression (cos 20°)(cos 30°) - (sin 20°)(sin 30°), we can use the trigonometric identity for the cosine of the difference of two angles:
cos(A - B) = cos A cos B + sin A sin B.
In this case, we can substitute A = 20° and B = 30°:
cos(20° - 30°) = cos 20° cos 30° + sin 20° sin 30°.
Now, let's evaluate cos(20° - 30°):
20° - 30° = -10°.
Since the cosine function is periodic, we can rewrite -10° as 350° (because 350° - 360° = -10°).
Therefore, we have:
cos(350°) = cos 20° cos 30° + sin 20° sin 30°.
Now, we need to evaluate cos 20°, cos 30°, sin 20°, and sin 30° separately:
cos 20° ≈ 0.9397,
cos 30° ≈ 0.8660,
sin 20° ≈ 0.3420,
sin 30° ≈ 0.5000.
Now substitute these values into the equation:
cos(350°) ≈ (0.9397)(0.8660) + (0.3420)(0.5000).
Compute the products separately:
(0.9397)(0.8660) ≈ 0.8132,
(0.3420)(0.5000) ≈ 0.1710.
Finally, substitute these values back into the equation:
cos(350°) ≈ 0.8132 + 0.1710.
Add the numbers:
cos(350°) ≈ 0.9842.
Therefore, the value of the expression (cos 20°)(cos 30°) - (sin 20°)(sin 30°) is approximately equal to 0.9842.
recall that cos(x+y) = cosx cosy - sinx siny
so, ...