Question

Evaluate each trigonometric expression to three significant digits. $$2.84 \cos 73.4^{\circ}-3.83 \tan 36.2^{\circ}$$

Short answer

The evaluated expression rounded to three significant digits is approximately -0.602.

Step-by-step solution

Calculate the cosine and tangent

Use a calculator or trigonometric tables to find the approximate values of the cosine of 73.4 degrees and the tangent of 36.2 degrees. Make sure the calculator is set to degree mode.

Multiply by the given coefficients

Multiply the cosine of 73.4 degrees by 2.84 and the tangent of 36.2 degrees by 3.83.

Subtract the results

Subtract the value found by multiplying the tangent of 36.2 degrees from the value found by multiplying the cosine of 73.4 degrees to find the answer.

Round the answer

Round the result to three significant digits as required by the exercise.

Key concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, providing relationships between the angles and sides of triangles, especially right-angled triangles. These functions include sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}), among others.

For instance, the cosine function relates the angle to the length of the adjacent side over the hypotenuse in a right-angled triangle. In contrast, the tangent function represents the ratio of the opposite side to the adjacent side. When evaluating expressions like \(2.84 \cos 73.4^\circ-3.83 \tan 36.2^\circ\), you're essentially combining these relationships to calculate a numerical value.

• \text{cos}\: The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• \text{tan}\: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

When solving problems involving trigonometric functions, accuracy and the correct usage of function properties are key to finding the correct solutions.

Significant Digits Rounding

Significant digits are critical in accurately representing the precision of a calculated value. Rounding to a specific number of significant digits ensures that the result of a calculation reflects the precision of the original data.

When asked to round to three significant digits, like in the given exercise \(2.84 \cos 73.4^\circ-3.83 \tan 36.2^\circ\), it implies that only the first three non-zero digits in the answer will be kept, and the rest will be rounded off. This process involves looking at the fourth digit to decide whether to round the third digit up or keep it as is.

For example, if a result comes out to 0.5684, when rounded to three significant digits, it would be 0.568. If that fourth digit were 5 or higher, you would round the third digit up to give 0.569.

Calculator Usage for Trigonometry

Calculators are indispensable tools for evaluating trigonometric expressions quickly and accurately. To employ a calculator effectively for trigonometry:

• Ensure your calculator is in the correct mode (degree or radian) based on the angle measurements in your problem. For angles measured in degrees, like in our case, the degree mode is necessary.
• Enter the angle into the calculator, followed by the corresponding trigonometric function key (\text{cos}, \text{sin}, \text{tan}).
• Record the calculated value and then multiply or add it according to the expression's requirements.

By carefully following these steps, you can minimize errors and verify the accuracy of your answers. Remember, calculators can differ in their functionalities, so familiarizing yourself with your specific model's instructions is crucial for success in solving trigonometric problems like \(2.84 \cos 73.4^\circ-3.83 \tan 36.2^\circ\).