Calculating Trigonometric Values Without a Calculator: cos 120° and Beyond

Understanding and calculating trigonometric values without the aid of a calculator or tables is a valuable skill. This article provides detailed methods and a step-by-step approach to finding the value of cos 120° and the problem of evaluating cos(120° - 45°). Additionally, we explore the Taylor series approximation for calculating cos 75°.

Calculating cos 120° Without a Calculator

To calculate cos 120° without using a table or calculator, we can use the properties of the unit circle and the cosine function.

Step-by-Step Process

Understand the Angle: The angle 120° is in the second quadrant of the unit circle. Reference Angle: The reference angle for 120° is found by subtracting it from 180°:?180° - 120° 60°. Cosine in the Second Quadrant: In the second quadrant, the cosine of an angle is negative. Hence, we have:

cos 120° -cos 60°

Value of cos 60°: The value of cos 60° is known to be:

cos 60° 1/2

Final Calculation: Now, substituting back:

cos 120° -cos 60° -1/2

Therefore: The value of cos 120° is -1/2.

An Alternative Argument:

Another way to calculate cos 120° is by using the identity that cos120° sin(90°-120°), which simplifies to cos120° -sin30° -1/2. This method uses the known trigonometric identity that cos x sin(90° - x) and sin(30°) 1/2.

Evaluating cos(120° - 45°) Without a Calculator

For evaluating cos(120° - 45°), we can first simplify the angle:

120° - 45° 75°

Now, we need to evaluate cos 75° without a calculator or table.

Taylor Series Approximation for cos 75°

Given that the cosine function is transcendental and cannot be expressed exactly in terms of the four basic arithmetic operations and roots, we can use the Taylor series approximation for cos x around 0. The first few terms of the Taylor series for cos x are:

cos x ≈ 1 - x^2/2! x^4/4! - x^6/6! ...

Step-by-Step Approximation

Reduction of Angle: Divide 75° by 2π and approximate it to a small value: Calculate the Argument: 75° ≈ 0.3984 radians (using π ≈ 3.1416). Apply the Taylor Series: Using the Taylor series for cos x ≈ 1 - x^2/2 x^4/24 - ... to approximate cos(0.3984): First Term: 1 Second Term: - (0.3984]^2 / 2 -0.07928 Third Term: 0.3984^4 / 24 ≈ 0.00105

Adding these terms together:

cos(0.3984) ≈ 1 - 0.07928 0.00105 0.92177

The actual value is cos 75° 0.9659, which when rounded to four decimal places matches our manually calculated value of 0.9218.

This method provided a good approximation while allowing us to perform manual calculations.