Question

Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$-11^{\circ} 18^{\prime}$$

Short answer

After converting to degrees, use a calculator in degree mode to find the values: sine is approximately equal to \text{-0.1960}, the cosine is approximately \text{0.9805}, and tangent is approximately \text{-0.2000} (to four significant digits).

Step-by-step solution

Convert Minutes to Degrees

Firstly, convert the angle from degrees and minutes to only degrees. For angular measures, 1 degree is equivalent to 60 minutes. To convert 18 minutes to degrees, divide 18 by 60: \(18 \div 60 = 0.3\). Now add this value to the degrees part of the angle, keeping in mind to maintain the negative sign: \(-11^\circ + 0.3^\circ = -11.3^\circ\).

Use a Calculator to Find Sine

Use a scientific calculator to find the sine of the angle \(-11.3^\circ\). Ensure the calculator is in degree mode, then input the angle and press the 'sine' function. Record the result to four significant digits.

Use a Calculator to Find Cosine

Similarly, use the scientific calculator to find the cosine of the angle \(-11.3^\circ\). Make sure the calculator is in degree mode, enter the angle, and press the 'cosine' function. Record the result to four significant digits.

Use a Calculator to Find Tangent

Finally, to find the tangent of the angle \(-11.3^\circ\), use the calculator again with the calculator still set to degree mode. Enter the angle and press the 'tangent' function. Record the result, but note if the tangent function is undefined.

Key concepts

Converting Minutes to Degrees

Understanding how to convert minutes to degrees is crucial when working with trigonometric functions because angles can be expressed not just in degrees but also in minutes and seconds. One degree is divided into 60 minutes ({60'}), and one minute is divided into 60 seconds ({60''}). When you encounter an angle in degrees and minutes, such as {-11^{{circ}} 18'}, the conversion to degrees only requires a simple division.

For instance, to convert the 18 minutes into degrees, you divide by 60:
{{18 {div} 60 = 0.3}}. Then, you add this to the degree part of the angle. If the degrees are negative, as in our example, we maintain the negative sign:
{{-11^{circ} + 0.3^{circ} = -11.3^{circ}}}. It's useful to perform this conversion before using a calculator to determine the sine, cosine, and tangent of the angle.

Calculating Sine Cosine Tangent

Once you have an angle solely in degrees, calculating its sine, cosine, and tangent is a breeze with a basic understanding of how these functions behave. Trigonometric functions evaluate the ratios of sides in a right-angled triangle corresponding to an angle.

To find the sine of an angle, we look at the ratio of the opposite side to the hypotenuse. The cosine is the adjacent side over the hypotenuse. The tangent, on the other hand, is the ratio of the opposite side to the adjacent side. These ratios are always the same for any given angle, no matter the size of the triangle. Hence, calculating these values for any angle boils down to using the correct function and ensuring that the angle is in the appropriate unit, which is degrees in this case.

Using Scientific Calculator

Using a scientific calculator simplifies the entire process of finding trigonometric function values. The key is to ensure that the calculator is set to the correct mode for angle measurements. In our case, this means degree mode, as angles can also be measured in radians or gradians.

To find the trigonometric values, one must input the angle after setting the calculator to degree mode. For sine, you'll hit the 'sine' function key; for cosine, the 'cosine' function key; and for tangent, the 'tangent' function key. The calculator does the work for you, and it's crucial to understand how to correctly read and enter the values to obtain accurate results. Always double-check the mode and syntax, especially while working with negative angles or converting between different units of angle measurement.

Trigonometric Function Values

Trigonometric function values, specifically sine, cosine, and tangent, are the numerical representations of the relationships between the sides of right-angled triangles. These values are critical in various fields including physics, engineering, and navigation. Sine and cosine values range from {-1} to {1}, while tangent values can range from negative to positive infinity, as the tangent function can approach infinity when the cosine value approaches zero.

It's also worth noting that certain angles have trigonometric function values that are well-known and often memorized, such as 0, 30, 45, 60, and 90 degrees. For other angles, a calculator is necessary to compute these values to the required precision. Keep in mind that the tangent function is undefined whenever the cosine of the angle is zero, which occurs at {90^{circ}} and its periodic intervals.