Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\tan 10^{\circ} \cot 10^{\circ}$$

Short answer

1

Step-by-step solution

Recall the identity for cotangent

Recall the identity \(\tan x \times \frac{1}{\tan x} = 1\). Since \(\tan x\) and \(\frac{1}{\tan x}\) are complementary functions, we use the fact that \(\tan x \times \text{cot} x = 1\).

Substitute the angle

Substitute \(x = 10^{\circ}\) into the identity. Thus, \(\tan 10^{\circ} \times \text{cot} 10^{\circ} = 1\).

Simplify the expression

Simplify the expression using the identity: \(\tan 10^{\circ} \text{cot} 10^{\circ} = 1\). Therefore, the exact value of the expression is 1.

Key concepts

cotangent

The cotangent of an angle is a trigonometric function that is the reciprocal of the tangent. In mathematical terms, for an angle \(\theta\), \(\text{cot} \theta = \frac{1}{\tan \theta}\).
This means that if you know the tangent of an angle, you can easily find its cotangent by taking the reciprocal.

The cotangent function is particularly useful in various trigonometric identities and equations.

• For example, if \(\tan 10^{\circ} = x\), then \(\text{cot} 10^{\circ} = \frac{1}{x}\).
• Since \(\text{cot} 10^{\circ} = \frac{1}{\tan 10^{\circ}}\), multiplying \(\tan 10^{\circ}\) by \(\text{cot} 10^{\circ}\) results in 1. This is because \(\tan 10^{\circ} \times \frac{1}{\tan 10^{\circ}}\) simplifies to 1.

tangent

The tangent of an angle in a right triangle is a basic trigonometric function. It is defined as the ratio of the opposite side to the adjacent side. For an angle \(\theta\): \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).

Tangent is widely used in various trigonometric identities and calculations. Understanding the tangent function helps in solving many trigonometric problems.

• For example, if the length of the side opposite to the angle \(\theta\) is 5 and the length of the adjacent side is 2, then \(\tan \theta = \frac{5}{2}\).
• This can be particularly useful in the context of complementary angles and their properties.

complementary angle

Complementary angles are two angles whose measures add up to 90 degrees. This concept is important in various trigonometric identities. When two angles are complementary, the sine of one angle is equal to the cosine of the other, and vice versa.

A key identity involving complementary angles is: \( \sin (90^{\circ} - \theta) = \cos \theta\) and \( \cos (90^{\circ} - \theta) = \sin \theta\).

• For example, if \( x = 10^{\circ}\) and \( y = 80^{\circ}\), then \( x + y = 90^{\circ} \), making them complementary angles.
• This relationship also holds for tangent and cotangent: \(\tan (90^{\circ} - \theta) = \cot \theta\) and \(\cot (90^{\circ} - \theta) = \tan \theta\).

fundamental identities

Fundamental trigonometric identities are the basic building blocks for solving trigonometric equations. These identities help in simplifying expressions and solving equations in trigonometry. Some of the most important identities include:

• Reciprocal identities, such as \(\tan \theta \times \text{cot} \theta = 1\).
• Pythagorean identities, like \(\sin^2 \theta + \cos^2 \theta = 1\).
• Quotient identities, such as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

By using these identities, you can simplify and solve complex trigonometric expressions. For instance, the exercise we looked at used the reciprocal identity to find that \(\tan 10^{\circ} \times \text{cot} 10^{\circ} = 1\).