Question

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \cot 70^{\circ} $$

Short answer

Approximately 0.36

Step-by-step solution

- Understand what the cotangent function represents

The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side. Mathematically, it is the reciprocal of the tangent function. Thus, \(\text{cot } \theta = \frac{1}{\text{tan } \theta}\).

- Convert degrees to radians if needed

Most calculators can work directly with degrees. If your calculator only works with radians, use the formula \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\). However, for this problem, leave the angle in degrees.

- Calculate the tangent of 70°

Using a calculator, find \(\text{tan } 70^{\text{ \circ}}\). Ensuring your calculator is set to degrees, input \(70\) and press the \(\text{tan}\) button. The calculator displays \(\text{tan } 70^{\text{ \circ}} \approx 2.747477419\).

- Find the reciprocal of the tangent value

The cotangent is the reciprocal of the tangent. Thus, calculate \(\text{cot } 70^{\text{ \circ}} \approx \frac{1}{2.747} \approx 0.363979\).

- Round to two decimal places

Finally, round the value \(\text{cot } 70^{\text{ \circ}} \approx 0.363979\) to two decimal places. Thus, the final answer is approximately \(\text{cot } 70^{\text{ \circ}} \approx 0.36\).

Key concepts

Trigonometric Functions

Trigonometric functions are essential in understanding relationships within right-angled triangles. They include sine, cosine, tangent, cotangent, secant, and cosecant.
Sine (sin) and cosine (cos) functions relate the lengths of the legs of a triangle to its hypotenuse. Tangent (tan) is the ratio of the opposite side to the adjacent side.
For instance, in a right-angled triangle, \[\text{tan } \theta = \frac{\text{opposite}}{\text{adjacent}}\].
Cotangent (cot) is the reciprocal of the tangent function. It relates the adjacent side to the opposite side.
Mathematically, \[\text{cot } \theta = \frac{1}{\text{tan } \theta}\].
Understanding these functions allows us to solve various problems in math, physics, and engineering by linking angles to side lengths.

Reciprocal

A reciprocal is the inverse of a number. For a non-zero number, its reciprocal is given by dividing 1 by that number.
For example:

• The reciprocal of 2 is \(\frac{1}{2}\)
• The reciprocal of 5 is \(\frac{1}{5}\)

In trigonometric functions, reciprocals help in transforming one function into another. The cotangent function is an excellent example of this.
Given \(\text{tan } \theta = x\), the cotangent function becomes \(\text{cot } \theta = \frac{1}{x}\).
This methodology simplifies calculations by switching between equivalent expressions and provides deeper insight into how angles and side lengths relate in triangles.

Degree to Radian Conversion

Angles can be measured in degrees or radians. Degrees are most commonly used in everyday language, while radians are more prevalent in higher mathematics.
To convert degrees to radians, use the formula:

• \[\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\text{π}}{180}\]

For example, to convert 70 degrees to radians:
\[70^{\text { \circ}} \times \frac{\text{π}}{180} = 1.22 \text{ rad}\]
While many calculators can handle degrees directly, understanding this conversion is crucial for more advanced math and science applications.
Knowing both radian and degree values of an angle ensures comprehensive insights into mathematical and physical problems.