Problem 72 - Algebra and Trigonometry

Question

Using a Calculator To solve a certain problem, you need to find the sine of 4 rad. Your study partner uses his calculator and tells you that $$ \sin 4=0.0697564737 $$ On your calculator you get $$ \sin 4=-0.7568024953 $$ What is wrong? What mistake did your partner make?

Step-by-Step Solution

Answer

The mistake was using degrees instead of radians for the angle measurement.

Step 1: Understand the Angle Measurement

First, recognize that calculators can work in either degrees or radians. In this problem, the angle '4' needs to be clarified whether it is in degrees or radians. Your calculator shows the result for radian measure, while your partner appears to have used the degree setting.

Step 2: Compare Sine Values

For a problem involving trigonometric functions, the calculator mode (degrees vs. radians) is critical. Let's compare the given values with commonly known sine values. Since \( \sin 4 \text{ radians} \approx -0.7568024953 \), and \( \sin 4 \text{ degrees} \approx 0.0697564737 \), this implies the discrepancy comes from differing angle measurements.

Step 3: Identify the Mistake

Determine from the above comparison that your partner made an error with the angle measurement. Instead of using radians, your partner calculated \( \sin 4\) degrees.

Step 4: Conclusion

Conclude that the correct interpretation for \( \sin 4\) {assuming the angle is in radian measure} is approximately \(-0.7568024953\), confirming that the error was due to the angle unit mismatch.

Key Concepts

Angle Measurement

When dealing with trigonometry, understanding angle measurement is crucial. Angles can be measured in two different units: degrees and radians. Knowing which one you are working with is important as they provide different values for trigonometric functions like sine, cosine, and tangent.

Calculators often allow you to switch between these measurements. However, using the wrong mode might lead to errors in calculations, as seen in the problem above. In the problem, the same angle '4' was interpreted differently: once in radians and once in degrees. This led to two different sine values, highlighting the importance of verifying your calculator's mode before starting your calculations. Always double-check whether you're in degrees or radians to avoid such mistakes.

Radians vs Degrees

Radians and degrees are two ways of expressing angles. Here’s a simple breakdown:

• Degrees divide a circle into 360 parts. It's a common unit in everyday life, like 90 degrees for a right angle.
• Radians use the radius of a circle. One complete circle (360 degrees) equals to 2π radians. Hence, π radians equals 180 degrees.

When switching between radians and degrees, remember:

• 180 degrees = π radians
• 1 degree = π/180 radians (approximately 0.01745 radians)
• 1 radian = 180/π degrees (approximately 57.2958 degrees)

Understanding this conversion is essential, especially when you switch the calculator's mode. For trigonometric functions, radians are often preferred in advanced mathematics and engineering because they create simpler derivatives and integrals. However, for everyday problems, degrees might be easier to understand. The key is to ensure consistency in usage.

Trigonometric Functions

Trigonometric functions are fundamental in exploring the relationships within triangles, especially right triangles. The basic functions are sine, cosine, and tangent.

• Sine (sin) of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the sine to the cosine, or the opposite side to the adjacent side.

In trigonometry, the angle measurement affects these values significantly. Hence, whether you're in degrees or radians changes the value output by your calculator.

For this reason, whenever working with these functions, remember to set your calculator to the appropriate unit (degrees or radians), based on the problem you're solving. Knowing how these functions work can aid in solving complex mathematical problems, but their correctness entirely depends on your understanding of the angle measurement involved.