Problem 21 - Algebra and Trigonometry

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

The other ratios are: \( \sin \theta = \cos \theta = \frac{\sqrt{2}}{2} \), \( \sec \theta = \csc \theta = \sqrt{2} \), \( \tan \theta = 1 \).

Step 1: Understand the given trigonometric ratio

We know that \( \cot \theta = 1 \), which means that the tangent of the angle \( \theta \) is the reciprocal of the cotangent, so \( \tan \theta = \frac{1}{1} = 1 \). This implies that the opposite side and the adjacent side of angle \( \theta \) are equal in length.

Step 2: Sketch the right triangle

Draw a right triangle with angle \( \theta \), such that the sides adjacent and opposite to \( \theta \) are equal. This forms a 45-45-90 triangle, which is an isosceles right triangle.

Step 3: Assign lengths to sides of the triangle

In a 45-45-90 triangle, the two legs are equal and the hypotenuse is \( \sqrt{2} \) times the length of one side. Assign both legs a length of 1, making the hypotenuse \( \sqrt{2} \).

Step 4: Calculate sine and cosine

Using the triangle, 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: Calculate secant and cosecant

Calculate the secant and cosecant using the reciprocal identities: \( \sec \theta = \frac{1}{\cos \theta} = \sqrt{2} \) and \( \csc \theta = \frac{1}{\sin \theta} = \sqrt{2} \).

Step 6: Summarize the trigonometric ratios

For \( \theta = 45^\circ \):- \( \sin \theta = \frac{\sqrt{2}}{2} \)- \( \cos \theta = \frac{\sqrt{2}}{2} \)- \( \tan \theta = 1 \)- \( \csc \theta = \sqrt{2} \)- \( \sec \theta = \sqrt{2} \)- \( \cot \theta = 1 \)

Key Concepts

Understanding Acute Angles

An acute angle is an angle that is less than 90 degrees. These angles are often found in everyday objects and are key concepts in geometry. When dealing with trigonometric ratios, acute angles are frequently part of the calculations, especially in right triangles. In the context of trigonometry, we focus on angles within the triangle that measure less than 90 degrees, typically ranging between 0 and 90 degrees. Understanding this concept is crucial when applying various trigonometric functions, such as sine, cosine, and tangent. These functions are often used to find relationships between sides and angles in a right triangle. When solving problems involving acute angles, remember that their sine and cosine values are always positive.

The Nature of Right Triangles

A right triangle is a type of triangle that has one angle measuring 90 degrees. The importance of this specific triangle lies in its geometric properties that allow for straightforward calculations using trigonometric ratios. In right triangles, the side opposite the right angle is known as the hypotenuse, which is the longest side. The other two sides are referred to as the 'legs' of the triangle.
This triangle is the basis for defining the trigonometric functions sine, cosine, and tangent, which are ratios comparing two sides of the triangle. Because of the 90-degree angle, calculations become more intuitive and are extensively applicable in various mathematical scenarios, including solving real-world problems.

Exploring Isosceles Right Triangles

An isosceles right triangle is a special type of right triangle where the two legs are of equal length. This results in two angles each measuring 45 degrees, making the angle opposite the equal sides also equal. The third angle, of course, is the right angle, measuring 90 degrees. In the context of trigonometry, the isosceles right triangle is useful because the relationship between its sides leads to straightforward calculations.

• Both legs are always equal, and if each leg measures 1, the hypotenuse will measure \( \sqrt{2} \).
• Trigonometric ratios for a 45-45-90 triangle are consistent, providing values like \( \sin \theta = \cos \theta = \frac{\sqrt{2}}{2} \) and \( \tan \theta = 1 \).

These properties make it an excellent tool for solving problems involving angles where the cotangent or tangent equals 1.

Understanding Reciprocal Identities

Reciprocal identities in trigonometry are relationships between the basic trigonometric functions and their inverted counterparts. Specifically, these identities allow us to express one trigonometric function in terms of another. Mastering these can enhance your ability to solve trigonometric problems effortlessly.

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

Understanding these identities helps in simplifying problems where calculation of particular angles or side lengths using trigonometric ratios is necessary. These concepts form the foundation for exploring more complex mathematical and geometric problems.