Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\sin 60^{\circ}=\frac{\sqrt{3}}{2}, \cos 60^{\circ}=\frac{1}{2}\) (a) \(\tan 60^{\circ}\) (b) \(\sin 30^{\circ}\) (c) \(\cos 30^{\circ}\) (d) \(\cot 60^{\circ}\)

Step-by-Step Solution

Answer

\(\tan 60^\circ = \sqrt{3}\), \(\sin 30^\circ = 1/2\), \(\cos 30^\circ = \sqrt{3}/2\), and \(\cot 60^\circ = \frac{1}{\sqrt{3}}\).

Step 1: Evaluate \(\tan 60^\circ\)

By using the identity, \(\tan x = \frac{\sin x}{\cos x}\), we can substitute \(\sin 60^\circ\) and \(\cos 60^\circ\) into the formula like this: \(\tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}\). So \(\tan 60^\circ = \sqrt{3}\).

Step 2: Evaluate \(\sin 30^\circ\)

The complementary nature of \(60^\circ\) and \(30^\circ\) means that \(\sin 30^\circ = \cos 60^\circ\). Thus, \(\sin 30^\circ = 1/2\).

Step 3: Evaluate \(\cos 30^\circ\)

Similarly, for \(\cos 30^\circ\), we use the complementary relationship and equate it to \(\sin 60^\circ\): \(\cos 30^\circ = \sin 60^\circ = \sqrt{3}/2\). Therefore, \(\cos 30^\circ = \sqrt{3}/2\).

Step 4: Evaluate \(\cot 60^\circ\)

For \(\cot 60^\circ\), we use the identity \(\cot x = \frac{1}{\tan x}\). Substituting in our previously found value for \(\tan 60^\circ\), we get \(\cot 60^\circ = \frac{1}{\sqrt{3}}\). Hence \(\cot 60^\circ = \frac{1}{\sqrt{3}}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental formulas that relate different trigonometric functions to one another. They are incredibly useful for evaluating trigonometric functions even when you don't have a calculator handy. These identities help identify relationships between angles and simplify complex equations.
a few key identities include:

• Pythagorean identities, like \( \sin^2 x + \cos^2 x = 1 \).
• Reciprocal identities, such as \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{1}{\tan x} \).
• Co-function identities, which explore relationships between complementary angles like \( \sin(90^\circ - x) = \cos x \).

Trigonometric identities are not just mathematical curiosities. They're practical tools that simplify computations and reveal deep mathematical insights.

Complementary Angles

Complementary angles are two angles whose measures add up to \(90^\circ\). In trigonometry, these angles have useful reciprocal properties, which can help simplify calculations.
When you know the trigonometric function of one angle, you can easily find the function of its complement. For example, knowing that \( \sin 60^\circ = \frac{\sqrt{3}}{2} \) helps you find that \( \sin 30^\circ = \cos 60^\circ = \frac{1}{2} \).
Similarly, these properties connect \( \cos 30^\circ \) and \( \sin 60^\circ \), where \( \cos 30^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \).
Understanding these complementary relationships can make solving trigonometric problems more intuitive and less reliant on memorization.

Tangent Function

The tangent function, noted as \( \tan \theta \), represents the ratio of the opposite side to the adjacent side in a right triangle. Mathematically, it's often defined with the identity \( \tan x = \frac{\sin x}{\cos x} \). This identity helps calculate the tangent of an angle when sine and cosine values are known.
Using this identity can quickly give you \( \tan 60^\circ \) when sine and cosine of \(60^\circ\) are given. Substituting these values, \( \tan 60^\circ = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \).
The tangent function is critical in solving many practical problems involving angles and slopes, like calculating heights or distances based on measured angles.

Cotangent Function

The cotangent function, \( \cot \theta \), is the reciprocal of the tangent function. Thus, it is expressed as \( \cot x = \frac{1}{\tan x} \). In terms of a right triangle, it corresponds to the ratio of the adjacent side to the opposite side.
This formula is particularly useful when you have the tangent of an angle and need to quickly derive the cotangent. For example, once \( \tan 60^\circ \) is known, finding \( \cot 60^\circ \) becomes straightforward by taking the reciprocal, resulting in \( \frac{1}{\sqrt{3}} \).
Cotangent, like tangent, often appears in problems dealing with angles' complementary and supplementary relationships or in optimizing certain geometric properties.