Problem 103 - Precalculus

Question

Calculate cot \(54.9^{\circ}\) in the following two ways: a. Find tan \(54.9^{\circ}\) to three decimal places and then divide 1 by that number. Write that number to five decimal places. b. With a calculator in degree mode, enter \(54.9,\) tan, \(1 / \mathrm{x},\) and round the result to five decimal places.

Step-by-Step Solution

Answer

a. 0.69930; b. 0.69930, both methods yield the same result.

Step 1: Calculate Tan 54.9°

Use a calculator to find \( \tan 54.9^{\circ} \). In degree mode, the calculator should show \( \tan 54.9^{\circ} \approx 1.430 \) rounded to three decimal places.

Step 2: Find Cotangent Using Tangent

To find \( \cot 54.9^{\circ} \), you divide 1 by \( \tan 54.9^{\circ} \). Thus, compute \( \frac{1}{1.430} \approx 0.69930 \). This is the cotangent value rounded to five decimal places.

Step 3: Use Reciprocal Function in Calculator

On a calculator, input 54.9 in degree mode. Press 'tan' to get \( \tan 54.9^{\circ} \approx 1.430 \). Then press '1/x' (the reciprocal function) to calculate \( \cot 54.9^{\circ} \). You should see \( \approx 0.69930 \) when rounded to five decimal places.

Key Concepts

Cotangent

The cotangent of an angle is a fundamental trigonometric function. It's the reciprocal of the tangent function. When you hear "cotangent," think, "the flip of tangent." If an angle has a tangent, we can figure out its cotangent by taking 1 and dividing it by the tangent value. This makes cotangent handy when dealing with triangles or circular functions, especially in trigonometric identities.
Understanding cotangent helps in various math applications like solving trigonometric equations or understanding wave patterns in physics.

• It's represented as \( ext{cot} \theta \).
• Formula: \( ext{cot} \theta = \frac{1}{ an \theta} \).
• For an angle \( \theta = 54.9^{\circ} \), if \( \tan 54.9^{\circ} \approx 1.430 \), then \( \text{cot} 54.9^{\circ} \approx 0.69930 \).

Tangent

Tangent is one of the primary trigonometric functions, often denoted as \( \tan \theta \). It tells us the ratio of the opposite side to the adjacent side in a right triangle. For angles in a circle, it describes the slope of a line from the origin.

• The tangent function is periodical. It repeats values every 180°.
• Unlike sine and cosine, which only range from -1 to 1, tangent values can be much greater or lower. This makes it pivotal for understanding curves and slopes.
• For example, at \( 54.9^{\circ} \), \( \tan 54.9^{\circ} \approx 1.430 \), meaning it's a decent positive slope.

Degree Mode

Degree mode on calculators is a setting that interprets angle measures in degrees rather than radians. When you're doing trigonometry on a calculator, it's crucial to use the correct mode.
In degree mode:

• Angles are measured in degrees, a full circle is 360 degrees.
• It's particularly useful in most educational settings in the U.S. where angles are commonly given in degrees.
• For example, when calculating \( \tan 54.9^{\circ} \), ensure your calculator is in degree mode for accurate results.

Using degree mode means getting results that make sense for angles described in the context of a circle split into 360 equal parts.

Calculator Use

Proper calculator use is essential when dealing with trigonometric functions. Calculators often have both a radian and degree mode; choosing the wrong one can drastically impact results.
When using a calculator for trigonometric calculations:

• Always double-check your calculator's mode before starting any calculations.
• Use functions in sequence: input an angle, use the trig function key (e.g., "tan"), and, if needed, the reciprocal ("1/x").
• Practicing these steps helps compute values like \( \cot 54.9^{\circ} \) efficiently, yielding \( \approx 0.69930 \).

Ensuring these practices will not only help in this particular calculation but will also build a solid foundation for tackling future trigonometry problems smoothly.