Question

Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$101^{\circ}$$

Short answer

For \(101^\circ\), the sine is approximately 0.9820, the cosine is approximately -0.1889, and the tangent is approximately -5.1446 (to four significant digits).

Step-by-step solution

Finding the sine of the angle

To find the sine of an angle of \(101^\circ\), use the sine function on a calculator. Make sure the calculator is set to degree mode and then enter the angle to get the value. The sine function is accessed by the \(\sin\) button.

Finding the cosine of the angle

To find the cosine of an angle of \(101^\circ\), use the cosine function on a calculator. Ensure the calculator is in degree mode, then press the \(\cos\) button and enter the angle to get the value.

Finding the tangent of the angle

To determine the tangent of an angle of \(101^\circ\), use the tangent function on the calculator. Make sure it's set to degree mode, press the \(\tan\) button, and then enter the angle. Since the tangent function can sometimes result in an undefined value for certain angles (namely those odd multiples of \(90^\circ\)), it's important to be aware of the possibility of an error.

Key concepts

Sine of an Angle

The sine of an angle in a right triangle is a fundamental concept in trigonometry. It is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. In simpler terms, if you have a right-angled triangle, and you're looking at one of the non-right angles, the sine of that angle tells you how 'tall' the triangle is relative to the longest side.

When using a calculator to find the sine of an angle like \(101^\circ\), you enter the angle after pressing the \(\sin\) button. Calculators typically have a button labeled with 'sin', which stands for sine. It's critical to ensure your calculator is in degree mode, not radian mode, as angles can be measured in both degrees and radians. For angle \(101^\circ\), the sine value gives a ratio, which after rounding to four significant digits can easily provide you the answer.

Cosine of an Angle

Cosine, often symbolized as cos, is another trigonometric function representing the ratio of the adjacent side to the hypotenuse of a right-angled triangle. Simply put, for any angle in a right-angled triangle, the cosine tells you how 'wide' the triangle is as compared to the hypotenuse.

The process of finding the cosine on a calculator is similar to that of the sine. After confirming that the calculator is in degree mode, the user presses the \(\cos\) button, inputs the degree of the angle, and the calculator computes the cosine. The ratio you get reflects how much the triangle stretches along the horizontal axis relative to the hypotenuse. This value can then be refined to four significant figures to give a precise answer.

Tangent of an Angle

The tangent function, abbreviated as tan, relates to the ratio of the opposite side over the adjacent side of a right-angled triangle. Unlike sine and cosine, it doesn't involve the hypotenuse. Tangent essentially measures the slope of the angle, or in practical terms, how steep the angle is.

When calculating the tangent for an angle like \(101^\circ\) using a calculator, you first ensure that the calculator is in degree mode. Then, by pressing the \(\tan\) button and entering the angle's degree, you're given the value of the tangent. Be mindful that for angles where the triangle is 'infinitely tall' (like at odd multiples of \(90^\circ\)), the tangent can be undefined because you'd be dividing by zero, leading to an error message.

Calculator Degree Mode

Calculators often come with various modes to measure angles—degrees and radians being the most common. The mode must match the unit of the angle you're working with. Degree mode is used for calculations involving angles measured in degrees.

To ensure accuracy in trigonometric calculations, it is imperative to switch the calculator to degree mode when working with degrees. This is usually done by pressing a mode button and selecting 'degree' as the unit. Failure to do so might yield incorrect values since trigonometric functions in radian mode will interpret the numbers differently.

Especially for students and professionals who switch between different units, double-checking the mode setting is a simple step that safeguards against easily avoidable mistakes.