(1 + tan 21 degrees)(1 + tan 28 degrees) * (1 + tan 24 degrees) * (1 + tan 17 degrees) = 4
if you google
(1 + tan 21°)(1 + tan 28°)(1 + tan 24°)(1 + tan 17°) = 4
you will find several proofs
Compound angel
To solve the equation (1 + tan 21 degrees)(1 + tan 28 degrees) * (1 + tan 24 degrees) * (1 + tan 17 degrees) = 4, we can break it down step by step.
Step 1: Convert degrees to radians
Since the trigonometric functions in standard calculators use radians, we need to convert the given angles from degrees to radians.
21 degrees = 21 * (π/180) radians
28 degrees = 28 * (π/180) radians
24 degrees = 24 * (π/180) radians
17 degrees = 17 * (π/180) radians
Step 2: Calculate the tangent of each angle
Using a calculator, find the tangent of each angle converted to radians:
tan(21 * (π/180)) ≈ 0.3746
tan(28 * (π/180)) ≈ 0.5317
tan(24 * (π/180)) ≈ 0.4450
tan(17 * (π/180)) ≈ 0.3057
Step 3: Substitute the tangent values into the equation
Replace the tangent values in the equation:
(1 + 0.3746)(1 + 0.5317)(1 + 0.4450)(1 + 0.3057) = 4
Step 4: Evaluate the equation
Calculate the expression:
(1 + 0.3746)(1 + 0.5317)(1 + 0.4450)(1 + 0.3057) ≈ 2.780
Therefore, the value of the expression is approximately 2.780, which is not equal to 4.
To solve the equation, we need to calculate the value of the expression (1 + tan 21 degrees)(1 + tan 28 degrees) * (1 + tan 24 degrees) * (1 + tan 17 degrees) step by step.
Step 1: Evaluate the tangent values.
The tangent function (tan) gives us the ratio of the opposite side to the adjacent side of a right triangle. However, we need to work with angles that are greater than 90 degrees, so we will use the periodicity of the tangent function to shift the angles to smaller values.
tan 21 degrees = tan (21 - 180) = tan (-159 degrees)
tan 28 degrees = tan (28 - 180) = tan (-152 degrees)
tan 24 degrees = tan (24 - 180) = tan (-156 degrees)
tan 17 degrees = tan (17 - 180) = tan (-163 degrees)
Step 2: Convert the negative angles to positive equivalents.
Since the tangent function is periodic, we can add or subtract multiples of 180 degrees to obtain the same value. Adding 180 degrees to a negative angle gives us the positive equivalent.
tan (-159 degrees) = tan (21 degrees)
tan (-152 degrees) = tan (28 degrees)
tan (-156 degrees) = tan (24 degrees)
tan (-163 degrees) = tan (17 degrees)
Step 3: Calculate the tangent values.
tan 21 degrees ≈ 0.391
tan 28 degrees ≈ 0.531
tan 24 degrees ≈ 0.455
tan 17 degrees ≈ 0.313
Step 4: Substitute the tangent values into the expression.
(1 + tan 21 degrees)(1 + tan 28 degrees) * (1 + tan 24 degrees) * (1 + tan 17 degrees)
≈ (1 + 0.391)(1 + 0.531) * (1 + 0.455) * (1 + 0.313)
≈ 1.391 * 1.531 * 1.455 * 1.313
≈ 4.019
Therefore, the value of the given expression is approximately 4.