Problem 43 - Basic Technical Mathematics with Calculus

Question

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\).$$\tan 70^{\circ}=\frac{\tan 30^{\circ}+\tan 40^{\circ}}{1-\left(\tan 30^{\circ}\right)\left(\tan 40^{\circ}\right)}$$

Step-by-Step Solution

Answer

The expression is verified: \( \tan 70^{\circ} = \frac{\tan 30^{\circ} + \tan 40^{\circ}}{1 - \tan 30^{\circ} \tan 40^{\circ}} \).

Step 1: Understanding the Relationship

The given expression is a tangent addition formula, which states \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). Here, we're asked to verify if \( \tan 70^{\circ} = \frac{\tan 30^{\circ} + \tan 40^{\circ}}{1 - \tan 30^{\circ} \tan 40^{\circ}} \).

Step 2: Calculate Individual Tangents

First, use a calculator to find \( \tan 30^{\circ} \) and \( \tan 40^{\circ} \), approximately: \( \tan 30^{\circ} \approx 0.5774 \) and \( \tan 40^{\circ} \approx 0.8391 \).

Step 3: Compute the Numerator

Calculate the numerator of the given fraction: \( \tan 30^{\circ} + \tan 40^{\circ} \approx 0.5774 + 0.8391 = 1.4165 \).

Step 4: Compute the Denominator

Calculate the denominator of the given fraction: \( 1 - \tan 30^{\circ} \tan 40^{\circ} \approx 1 - (0.5774 \times 0.8391) = 1 - 0.4844 = 0.5156 \).

Step 5: Evaluate the Entire Expression

Evaluate the fraction \( \frac{1.4165}{0.5156} \approx 2.7475 \).

Step 6: Compare with \( \tan 70^{\circ} \)

Finally, use a calculator to verify \( \tan 70^{\circ} \approx 2.7475 \). The values match, confirming the given identity.

Key Concepts

Understanding the Tangent Addition Formula

In trigonometry, various identities provide relationships between angles and their respective trigonometric ratios.The tangent addition formula is particularly useful for understanding how two angles add together in terms of their tangent values.It is given by:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

This formula allows you to find the tangent of the sum of two angles by using the individual tangent values of those angles.In the exercise, we verify if the equation \( \tan 70^{\circ} = \frac{\tan 30^{\circ} + \tan 40^{\circ}}{1 - \tan 30^{\circ} \tan 40^{\circ}} \) holds true.You can see it follows the structure of the tangent addition formula, setting \( A = 30^{\circ} \) and \( B = 40^{\circ} \).Thus, knowing the addition formula helps you easily compare and verify whether such equations are identities.

Effectively Using Calculators for Trigonometric Calculations

When working with angles and their trigonometric functions, a calculator can be an invaluable tool.Calculators quickly provide you accurate results for trigonometric values, such as tangent, sine, and cosine.In our exercise, the use of a calculator helps us calculate \( \tan 30^{\circ} \approx 0.5774 \) and \( \tan 40^{\circ} \approx 0.8391 \). These provide precise values necessary for accurate computation.Using a calculator not only saves time but also helps minimize human errors during manual calculations.To effectively use your calculator:

• Ensure it is set in the correct mode—usually degrees or radians—based on the context.
• Enter the angle values correctly and double-check the mode before calculating.
• Have your calculator handy when evaluating expressions with multiple trigonometric terms.

Verifying Trigonometric Expressions

Verifying trigonometric identities or expressions involves proving that one side of the equation can be manipulated to become identical to the other side.In our context, we start with the statement \( \tan 70^{\circ} = \frac{\tan 30^{\circ} + \tan 40^{\circ}}{1 - \tan 30^{\circ} \tan 40^{\circ}} \) and aim to confirm its truth.This involves several steps:

• Calculate: First obtain the exact or approximate values of \( \tan 30^{\circ} \) and \( \tan 40^{\circ} \).
• Evaluate: Insert these values into the formula and calculate the result, checking both numerator and denominator values carefully.
• Compare: Finally, evaluate \( \tan 70^{\circ} \) separately and compare the results from both sides of the identity.

In the exercise, we end up confirming that \( \tan 70^{\circ} \approx 2.7475 \) matches the calculated expression value.The matching results establish the expression's validity, demonstrating the skill of verifying trigonometric identities as crucial in understanding and applying trigonometry basics.