Problem 42 - Basic Technical Mathematics with Calculus

Question

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\).$$\sin ^{2} 77.5^{\circ}+\cos ^{2} 77.5^{\circ}=1$$

Step-by-Step Solution

Answer

The identity holds: \(\sin^2 77.5^\circ + \cos^2 77.5^\circ = 1\).

Step 1: Understand the Equation

The given equation is an identity that comes from the Pythagorean trigonometric identity, which states that for any angle \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\). We need to verify this identity numerically for \(\theta = 77.5^\circ\).

Step 2: Calculate \(\sin 77.5^\circ\)

Use a calculator to find the sine of 77.5 degrees. Ensure your calculator is set to degree mode, and you should find that \(\sin 77.5^\circ \approx 0.97437\).

Step 3: Square the Sine Value

Calculate \((\sin 77.5^\circ)^2\) using the value obtained in Step 2. This would be approximately \((0.97437)^2 \approx 0.94940\).

Step 4: Calculate \(\cos 77.5^\circ\)

Use a calculator to find the cosine of 77.5 degrees. With the calculator in degree mode, you should find that \(\cos 77.5^\circ \approx 0.22495\).

Step 5: Square the Cosine Value

Calculate \((\cos 77.5^\circ)^2\) using the value obtained in Step 4. This would be approximately \((0.22495)^2 \approx 0.05060\).

Step 6: Add the Squared Values

Add the squared sine and cosine values obtained in Steps 3 and 5. Check if the sum is approximately equal to 1: \(0.94940 + 0.05060 \approx 1.00000\).

Step 7: Verify the Identity

Since the sum \(0.94940 + 0.05060\) is approximately equal to 1, this verifies the given trigonometric identity for \(\theta = 77.5^\circ\). The slight differences are due to the rounding of decimal places in calculations.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It links the squared values of sine and cosine for any angle \(\theta\) to the constant 1. This identity is expressed as:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This equation is closely related to the Pythagorean theorem. Imagine a right triangle on the unit circle (a circle with radius 1). Here, the x-coordinate is \(\cos \theta\), and the y-coordinate is \(\sin \theta\). By the Pythagorean theorem, the sum of the squares of these coordinates equals the square of the hypotenuse, which is 1 in the unit circle. Thus, the identity is always true regardless of the specific angle.
To see the identity in action, consider an angle like \(77.5^{\circ}\). By calculating the sine and cosine for this angle, and squaring each, the sum remains 1.

Sine and Cosine Functions

Sine and cosine are two of the most important trigonometric functions, representing projections of a point on the unit circle.
They describe how the x and y coordinates of a rotating radius around a circle change.

• The sine function \(\sin \theta\) gives the y-component (vertical) of the triangle.
• The cosine function \(\cos \theta\) provides the x-component (horizontal) of the triangle.

For every angle \(\theta\) in degrees, these functions return a value between -1 and 1. When combined with the Pythagorean identity, they demonstrate a clear relationship between these coordinate projections.
The calculations in earlier steps were approximations for \(\theta = 77.5^{\circ}\), resulting in:

• \(\sin 77.5^{\circ} \approx 0.97437\)
• \(\cos 77.5^{\circ} \approx 0.22495\)

Despite the specific angle, the Pythagorean identity still holds, showcasing the beauty of trigonometric relationships.

Angle Measurement in Degrees

Degrees are a unit of measurement used to specify the size of an angle. Typically, a full circle is divided into 360 degrees. When tackling trigonometric problems, knowing whether your calculator is set to degrees or radians is crucial.

• In the exercise, the angle \(77.5^{\circ}\) is discussed.
• This means calculations were done assuming a degree-based system.

Compared to the radian measure, which is a more mathematical-oriented unit depending on \(\pi\), degrees offer a more intuitive approach. For practical applications and teaching, degrees are often preferred due to their simplicity in visualization and usage.
Switching your calculator to degree mode, as highlighted in the solution, ensures accurate results while working with actions and conversions involving degree angles.