Problem 98 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. $$\text { Show that } \tan \left(90^{\circ}-\theta\right)=\cot \theta$$

Step-by-Step Solution

Answer

Proved. The given equation \( \tan \left(90^{\circ}-\theta\right)=\cot \theta \) is indeed a recognized trigonometric identity. Therefore, it holds true.

Step 1: Understand trigonometric properties

Mainly, we need to recall two basic trigonometric properties: 1) \( \tan (90^{\circ} - \theta) = \cot \theta \) 2) \( \cot \theta = 1 / \tan \theta \). These two properties are key to solving the equation.

Step 2: Manipulate the given equation

Begin with the equation which is provided. We have \( \tan (90^{\circ} - \theta) \). According to the trigonometric property that \( \tan (90^{\circ} - \theta) = \cot \theta \), so \( \tan (90^{\circ} - \theta) = \cot \theta \) holds.

Step 3: Conclude the proof

From Step 2, we found out that \( \tan (90^{\circ} - \theta) = \cot \theta \). This is exactly what we were asked to prove! Therefore, we have completed our proof.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships involving trigonometric functions. These are mathematical equations that are true for all values of the variables where both sides of the equation are defined.

In our exercise, we deal with a trigonometric identity involving the tangent and cotangent functions. Specifically, we are interested in the identity \( \tan(90^{\circ} - \theta) = \cot \theta \).

Let's break it down a bit:

• \( \tan \theta \) is the ratio of the opposite side to the adjacent side in a right triangle.
• \( \cot \theta \) is the reciprocal of \( \tan \theta \), meaning \( \cot \theta = 1/\tan \theta \).
• The identity \( \tan(90^{\circ} - \theta) = \cot \theta \) is a result of complementary angles, which states that the sine of an angle is equal to the cosine of its complement and vice versa.

These identities are instrumental in solving various problems in trigonometry, allowing for the simplification of expressions and solving equations.

Cotangent Function

The cotangent function, \( \cot \theta \), is one of the primary trigonometric functions. It is defined as the reciprocal of the tangent function:
\[ \cot \theta = \frac{1}{\tan \theta} \]

Understanding the cotangent function begins with knowing its geometric interpretation. It is the ratio of the adjacent side to the opposite side in a right triangle. This means:

• When \( \theta \) is an acute angle in a right triangle, \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \).
• It is defined for all angles except where \( \tan \theta = 0 \), since division by zero is undefined.
• As \( \theta \) approaches \( 0^{\circ} \) or \( 180^{\circ} \), \( \cot \theta \) can be very large (positive or negative depending on direction).

In the context of identities, realizing that the cotangent is merely the flip of the tangent helps in visualizing and verifying identities that involve this function.

Proof Techniques

Proof techniques in mathematics provide structured ways to show that a statement or identity holds true in all relevant scenarios. In trigonometry, these methods are especially crucial, given the often complex relationships between angles and functions.

Let's explore some techniques used to derive or verify trigonometrical identities, such as \( \tan(90^{\circ} - \theta) = \cot \theta \):

• Understanding Basic Definitions: Begin by revisiting the definitions of trigonometric functions like sine, cosine, tangent, and cotangent. Knowing these helps in transforming one function into another.
• Using Known Identities: Utilize established identities such as Pythagorean identities or complementary angle identities. They often serve as shortcuts in proving new ones.
• Substitution: If an equation is given, replace parts of the expression with equivalent terms from known identities or definitions. This can simplify the expression greatly.
• Simplification: Focus on transforming the complex side of the equation into the simpler side using algebraic manipulations.

Proof techniques offer the clarity needed to resolve seemingly complex identities, turning them into straightforward, logical steps that demonstrate their truth.