Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate to three decimal places. (a) \(\tan 21^{\circ} 10^{r}\) (b) \(\cot 1.13\)

Step-by-Step Solution

Answer

(a) 0.387 (b) 0.410

Step 1: Convert the Angle to Decimal Degrees

Convert the angle from degrees and minutes to decimal degrees. Given 21°10', convert to decimal degrees as follows: \[ 21^{\circ} 10^{'} = 21 + \frac{10}{60} = 21.1667^{\circ} \].

Step 2: Calculate the Tangent

Use a calculator to find the tangent of the angle in decimal degrees. For angle 21.1667°, calculate \( \tan 21.1667^{\circ} \approx 0.387 \).

Step 3: Calculate the Cotangent

Find the cotangent of the given angle in radians. For \( \cot 1.13 \), use the relationship \[ \cot(\theta) = \frac{1}{\tan(\theta)} \].Calculate \( \tan(1.13) \approx 2.442 \), hence \( \cot(1.13) \approx \frac{1}{2.442} \approx 0.410 \).

Key Concepts

Tangent

Tangent is one of the fundamental trigonometric functions. It is usually abbreviated as "tan". It is defined using a right triangle as the ratio of the length of the opposite side to the length of the adjacent side for a given angle in the triangle. In symbols, it's written as

• \( an(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

Tangent can also be represented on the unit circle as the y-coordinate divided by the x-coordinate, which gives the same ratio as in the right triangle but extends the concept to any angle, not just those in a triangle.

When manipulating tangent for calculations, we often need the tangent of angles in decimal fractions of degrees, or in radians. Calculators can easily compute \(\tan\) for both. It is important to make sure that your calculator is set to the appropriate unit of measure, either degrees or radians, to avoid errors.

Cotangent

Cotangent is another essential trigonometric function. It's the reciprocal of the tangent function. If tangent describes the ratio of the opposite side over the adjacent side in a right triangle, cotangent reverses that ratio. It states that:

• \(\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan(\theta)}\)

This reciprocal relationship is vital because it means that once you know the tangent of an angle, you can quickly find the cotangent simply by flipping that value. Cotangent can be less intuitive to compute directly, so understanding its relationship with tangent makes solving problems involving cotangent much easier.

In angle circles, where angles go beyond the right triangle limit, cotangent can help solve complex trigonometric problems. Just ensure to maintain the correct relationship with tangent, especially when calculating with radians and degrees.

Radians

Radians are another way to measure angles, different from degrees. While a full circle is 360 degrees, it is also \(2\pi\) radians. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius.

• To understand this, recall that the circumference of a circle is \(2\pi\times\text{radius}\), so the full angle is \(2\pi\) radians.
• This means that one radian is approximately 57.2958 degrees.

Using radians instead of degrees can simplify many mathematical equations, particularly in calculus and engineering contexts. Most scientific calculators allow you to switch between degrees and radians, permitting easy access to either as needed.

Keep in mind that when calculating trigonometric values in radians, make sure all values remain in this unit to prevent miscalculations.

Angle Conversion

Converting angles between degrees and radians is a common task in trigonometry and is straightforward. The relationship between degrees and radians is derived from the equivalent of \(360^{\circ}\) being \(2\pi\) radians.

• To convert from degrees to radians, use the formula: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
• Conversely, to convert from radians to degrees, use: \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\).

This conversion is critical in ensuring that trigonometric calculations yield correct results, especially when using functions like tangent and cotangent in different systems that may default to degrees or radians without your awareness.

Always verify the unit requirements in the problem, and make a habit of converting angles when necessary to avoid unexpected results.