Problem 25 - Precalculus : Building Concepts and Connections

Question

Use the given value of a trigonometric function of \(\theta\) to find the values of the other five trigonometric functions. Assume \(\theta\) is an acute angle. $$\cot \theta=4$$

Step-by-Step Solution

Answer

The corresponding trigonometric function values for θ are: \(\sin \theta = \frac{1}{\sqrt{17}}\), \(\cos \theta = \frac{4}{\sqrt{17}}\), \(\tan \theta = \frac{1}{4}\), \(\sec \theta = \frac{\sqrt{17}}{4}\), \(\csc \theta = \sqrt{17}\)

Step 1: Identify given function value

Start by understanding that we are given \(\cot \theta = 4\), indicating a cotangent of a right angle. This function represents the ratio of adjacent side to opposite side in a right triangle.

Step 2: Define a right triangle

From \(\cot \theta = 4\), we can construe a right triangle where the adjacent side is 4 units and the opposite side is 1 unit. This is because cotangent is the reciprocal of tangent and the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side that forms the angle.

Step 3: Calculate the hypotenuse

Utilize the Pythagorean theorem \(a^2 + b^2 = c^2\), where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse. Thus, the hypotenuse is \( \sqrt{1^2 + 4^2} = \sqrt{17}\).

Step 4: Calculate the remaining trigonometric functions

Using the lengths of the sides of the triangle, we can find the values of the other trigonometric functions. The sine of θ is given by the ratio of the opposite side to the hypotenuse, so \(\sin \theta = \frac{1}{\sqrt{17}}\). The cosine of θ is given by the ratio of the adjacent side to the hypotenuse, so \(\cos \theta = \frac{4}{\sqrt{17}}\). The tangent of θ is the ratio of the opposite side to the adjacent side, so \(\tan \theta = \frac{1}{4}\). Finally, the secant and cosecant are defined as the reciprocals of cosine and sine respectively, so \(\sec \theta = \frac{\sqrt{17}}{4}\) and \(\csc \theta = \sqrt{17}\).

Key Concepts

Right Triangle

A right triangle is a triangle that has one angle measuring exactly 90 degrees. In such triangles, the side opposite the right angle is called the hypotenuse. This is the longest side of the triangle. The other two sides are referred to as the legs, which form the right angle.
The angles in a right triangle add up to 180 degrees, with one angle being 90 degrees. This means the sum of the other two angles is also 90 degrees. Thus, these two angles are always acute, meaning each is less than 90 degrees.

• Right angle: 90 degrees
• Opposite: Side opposite the angle of interest
• Adjacent: Side adjacent to the angle of interest

Each side of the triangle is part of forming the trigonometric functions.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle established by the ancient Greek mathematician Pythagoras. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
In the context of the example, given the adjacent side length of 4 units and opposite side length of 1 unit, the Pythagorean Theorem is used to find the hypotenuse. Calculating, we find \( c = \sqrt{1^2 + 4^2} = \sqrt{17} \).

• Ensures the triangle is right-angled.
• Helps calculate unknown side lengths.

The theorem forms the basis for solving problems related to right triangles.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. Each function corresponds to one part of the triangle’s sides and angles. In the given exercise, we're focusing on the six primary trigonometric functions:

• **Sine (sin)**: Ratio of the opposite side to the hypotenuse. For \( \theta \), \( \sin \theta = \frac{1}{\sqrt{17}} \).
• **Cosine (cos)**: Ratio of the adjacent side to the hypotenuse. \( \cos \theta = \frac{4}{\sqrt{17}} \).
• **Tangent (tan)**: Ratio of the opposite side to the adjacent side. \( \tan \theta = \frac{1}{4} \).
• **Cotangent (cot)**: Reciprocal of tan. Given as \(4\).
• **Secant (sec)**: Reciprocal of cos. \( \sec \theta = \frac{\sqrt{17}}{4} \).
• **Cosecant (csc)**: Reciprocal of sin. \( \csc \theta = \sqrt{17} \).

These functions are tools to explore angle-side relationships and solve problems involving right triangles.