Problem 21 - Precalculus Mathematics for Calculus

Question

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\). $$\cot \theta=1$$

Step-by-Step Solution

Answer

\(\sin \theta = \frac{\sqrt{2}}{2}, \cos \theta = \frac{\sqrt{2}}{2}, \tan \theta = 1, \csc \theta = \sqrt{2}, \sec \theta = \sqrt{2}\), and \(\cot \theta = 1\).

Step 1: Understand the Given Information

We know that \(\cot \theta = 1\). Since \(\cot \theta = \frac{1}{\tan \theta}\), it implies that \(\tan \theta = 1\).

Step 2: Identify the Right Triangle Configuration

A triangle where \(\tan \theta = 1\) indicates that the opposite side and the adjacent side are of equal length (because \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)). One such right triangle is the isosceles right triangle (45°-45°-90° triangle), where both the legs are equal and angles are 45° each.

Step 3: Calculate the Hypotenuse

For an isosceles right triangle where each leg is 1, the hypotenuse can be found using the Pythagorean theorem: \[ c = \sqrt{1^2 + 1^2} = \sqrt{2} \]

Step 4: Determine Sine and Cosine Ratios

Using the legs and hypotenuse, calculate:\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\), and \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\).

Step 5: Determine Secant and Cosecant Ratios

Use the reciprocal identities: \(\csc \theta = \frac{1}{\sin \theta} = \sqrt{2}\), and \(\sec \theta = \frac{1}{\cos \theta} = \sqrt{2}\).

Step 6: Verifying the Tangent Ratio Again

\(\tan \theta\) was given as 1, which matches our earlier identification of the triangle as an isosceles right triangle. Thus, all calculations align with the given \(\cot \theta = 1\).

Key Concepts

Right Triangle

In trigonometry, a right triangle is fundamental. It is a triangle with one angle measuring exactly 90 degrees. The essence of studying these triangles lies within their sides:

• Legs: These are the two shorter sides adjacent to the right angle.
• Hypotenuse: This is the longest side opposite the right angle.

Right triangles are pivotal when dealing with trigonometric functions, revealing relationships within the angles and sides. Understanding their properties offers a foundation for comprehending more complex trigonometric applications.

Isosceles Right Triangle

An isosceles right triangle is a special kind of right triangle. As its name suggests, this triangle has two equal sides (the legs) and a right angle. It also features two equal angles, both measuring 45 degrees.

This specific configuration simplifies calculations, since:

• When one leg is equal to 1 unit, the other leg will also be 1 unit.
• The hypotenuse, using the Pythagorean theorem, is \[ \sqrt{2} \].

These attributes make the isosceles right triangle a convenient model for determining exact trigonometric ratios of 45-degree angles.

Sine and Cosine

In trigonometry, sine and cosine are two essential ratios that relate an angle to the lengths of sides in a right triangle.
For an angle \( \theta \):

• The sine function (\( \sin \theta \)) is the ratio of the opposite side to the hypotenuse.
• The cosine function (\( \cos \theta \)) is the ratio of the adjacent side to the hypotenuse.

For the isosceles right triangle with legs each being 1 and hypotenuse \( \sqrt{2} \):

• \( \sin \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)
• \( \cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)

Knowing these ratios plays an important role in understanding how the angles govern the side ratios in right triangles.

Pythagorean Theorem

The Pythagorean theorem is a powerful tool in trigonometry, guiding us to solve right triangle problems effortlessly. It is mathematically expressed as:
\[a^2 + b^2 = c^2\]where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse of a right triangle.

For our isosceles right triangle example:

• Taking each leg \( a = b = 1 \),
• The hypotenuse \( c = \sqrt{1^2 + 1^2} = \sqrt{2} \).

This theorem helps to verify if a triangle is right, ensuring the side lengths correspond to this equation. It also lays the groundwork for calculating further trigonometric identities and ratios.

Reciprocal Identities

Reciprocal identities in trigonometry offer an alternate perspective on basic trigonometric functions. They relate the primary functions (sine, cosine, and tangent) to other functions:

• The cosecant of \( \theta \) (\( \csc \theta \)) is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• The secant of \( \theta \) (\( \sec \theta \)) is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• The cotangent of \( \theta \) (\( \cot \theta \)) is the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \).

Using these identities, we expand our toolkit for solving trigonometric equations, especially when one trigonometric ratio is known, such as \( \cot \theta = 1 \) in this exercise example. They provide a way to derive values that complement the primary trigonometric functions.