Problem 119 - A Graphical Approach to Precalculus with Limits

Question

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\cot \theta=\frac{\sqrt{3}}{8},\) given that \(\theta\) is in quadrant I

Step-by-Step Solution

Answer

\(\sin \theta = \frac{8\sqrt{67}}{67}, \cos \theta = \frac{\sqrt{3} \sqrt{67}}{67}, \tan \theta = \frac{8\sqrt{3}}{3}\), etc.

Step 1: Understand the cotangent function

The cotangent of an angle \(\theta\) in a right triangle is defined as \(\cot \theta = \frac{\text{adjacent side}}{\text{opposite side}}\). In this problem, \(\cot \theta = \frac{\sqrt{3}}{8}\), which means the adjacent side is \(\sqrt{3}\) and the opposite side is \(8\).

Step 2: Find the hypotenuse using the Pythagorean theorem

Use the Pythagorean theorem, \(c^2 = a^2 + b^2\), to find the hypotenuse \(c\). Here, \(a = \sqrt{3}\) and \(b = 8\). Thus, \[c = \sqrt{(\sqrt{3})^2 + 8^2} = \sqrt{3 + 64} = \sqrt{67}.\]

Step 3: Find the sine of \(\theta\)

The sine function is defined as \(\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\). So, \(\sin \theta = \frac{8}{\sqrt{67}}\). Simplify this to get \(\sin \theta = \frac{8\sqrt{67}}{67}\).

Step 4: Find the cosine of \(\theta\)

The cosine function is defined as \(\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\). So, \(\cos \theta = \frac{\sqrt{3}}{\sqrt{67}}\). Simplify this to get \(\cos \theta = \frac{\sqrt{201}}{67}\).

Step 5: Find the tangent of \(\theta\)

The tangent function is the reciprocal of the cotangent function. \(\tan \theta = \frac{1}{\cot \theta} = \frac{8}{\sqrt{3}}\). Rationalize the denominator to get \(\tan \theta = \frac{8\sqrt{3}}{3}\).

Step 6: Find the secant of \(\theta\)

The secant function is the reciprocal of the cosine function. \(\sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{67}}{\sqrt{3}}\). Rationalize the denominator to get \(\sec \theta = \frac{\sqrt{201}}{3}\).

Step 7: Find the cosecant of \(\theta\)

The cosecant function is the reciprocal of the sine function. \(\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{67}}{8}\).

Step 8: Compile all trigonometric functions of \(\theta\)

Now, we have all the trigonometric functions:- \(\sin \theta = \frac{8\sqrt{67}}{67}\)- \(\cos \theta = \frac{\sqrt{3} \sqrt{67}}{67}\)- \(\tan \theta = \frac{8\sqrt{3}}{3}\)- \(\cot \theta = \frac{\sqrt{3}}{8}\)- \(\sec \theta = \frac{\sqrt{201}}{3}\)- \(\csc \theta = \frac{\sqrt{67}}{8}\)

Key Concepts

Cotangent Function

The cotangent function is one of the basic trigonometric functions, which is particularly useful in right-angled triangles. When we talk about \(\cot \theta\), we are referring to the ratio of the length of the adjacent side to the length of the opposite side in a right triangle. This can be expressed as:

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

Understanding this function is crucial as it helps in finding various other trigonometric values. In our given problem, \(\cot \theta = \frac{\sqrt{3}}{8}\). This implies that for the angle \(\theta\), if the adjacent side of a triangle is taken to be \(\sqrt{3}\), then the opposite side is 8 units. Remembering that cotangent is the reciprocal of the tangent function will always help in switching between these functions easily.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly in relation to right-angled triangles. The theorem states that in a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides. Mathematically expressed as:

• \( c^2 = a^2 + b^2 \)

In the context of our problem, this theorem is used to calculate the length of the hypotenuse when the lengths of the other two sides are known. Here we have \(a = \sqrt{3}\) and \(b = 8\). Plugging these into the theorem gives us:

• \( c = \sqrt{\left(\sqrt{3}\right)^2 + 8^2} = \sqrt{3 + 64} = \sqrt{67} \)

This hypotenuse \(\sqrt{67}\) is crucial for determining the values of other trigonometric functions like sine and cosine.

Trigonometric Ratios

Trigonometric ratios are the relationships between the lengths of sides in a right triangle relative to one of its acute angles. There are six primary trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent, each serving a different purpose. Here, each function is a ratio:

• Sine (\(\sin \theta\)): The ratio of the opposite side to the hypotenuse, \(\frac{8}{\sqrt{67}} = \frac{8\sqrt{67}}{67}\).
• Cosine (\(\cos \theta\)): Derived from the ratio of the adjacent side to the hypotenuse, \(\frac{\sqrt{3}}{\sqrt{67}} = \frac{\sqrt{3}\sqrt{67}}{67}\).
• Tangent (\(\tan \theta\)): The ratio of the opposite side to the adjacent side, computed as the reciprocal of cotangent, \(\frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3}\).

Understanding these ratios allows us to solve various problems in geometry and physics. Once you've calculated the primary functions, others like secant, cosecant, and cotangent simply require knowing their reciprocal or original function.