Problem 31 - Precalculus: A Concise Course

Question

Use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. \(\sin 60^{\circ}=\frac{\sqrt{3}}{2}, \quad \cos 60^{\circ}=\frac{1}{2}\) (a) \(\sin 30^{\circ}\) (b) \(\cos 30^{\circ}\) (c) \(\tan 60^{\circ}\) (d) \(\cot 60^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Finding \(\sin 30^{\circ}\)

By using the cofunction identity of sine, which is \(\sin (90^{\circ} - \theta) = \cos \theta\), we substitute \(\theta\) with \(60^{\circ}\) to find that \(\sin 30^{\circ} = \cos 60^{\circ}\). Given that the value of \(\cos 60^{\circ}\) is \(\frac{1}{2}\), we find that \(\sin 30^{\circ} = \frac{1}{2}\).

Step 2: Finding \(\cos 30^{\circ}\)

Using the cofunction identity for cosine, which is \(\cos (90^{\circ} - \theta) = \sin \theta\), we substitute \(\theta\) with \(60^{\circ}\). This gives us \(\cos 30^{\circ} = \sin 60^{\circ}\). Given the function value of \(\sin 60^{\circ}\) which is \(\frac{\sqrt{3}}{2}\), we find that \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).

Step 3: Finding \(\tan 60^{\circ}\)

The tangent function can be defined as the ratio of sine to cosine, thus \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the given function values of \(\sin 60^{\circ}\) and \(\cos 60^{\circ}\), we find that \(\tan 60^{\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}\).

Step 4: Finding \(\cot 60^{\circ}\)

The cotangent function is the reciprocal of the tangent function, thus \(\cot \theta = \frac{1}{\tan \theta}\). Using the previously calculated value of \(\tan 60^{\circ}\), we find that \(\cot 60^{\circ} = \frac{1}{\sqrt{3}}\).

Key Concepts

Cofunction Identities

Cofunction identities are a key concept in trigonometry that relate the trigonometric functions of complementary angles. Complementary angles are two angles that add up to 90 degrees. The identities are essential for simplifying and solving problems involving right triangles. For example,

• The cofunction identity for sine is \( \sin(90^{\circ} - \theta) = \cos \theta \), which relates the sine and cosine of complementary angles.
• Similarly, for cosine, \( \cos(90^{\circ} - \theta) = \sin \theta \), allowing you to express one function in terms of another.

Understanding these identities enables quick computation of trigonometric values for angles like 30°, 60°, and 90°. For instance, using the given value of \( \cos 60^{\circ} = \frac{1}{2} \), applying the sine cofunction identity helps to confirm that \( \sin 30^{\circ} = \frac{1}{2} \), illustrating how cofunction identities simplify calculations.

Sine and Cosine Values

In trigonometry, grasping the values of sine and cosine for specific angles is pivotal. These values are basic and often used in more elaborate computations. For standard angles like 30°, 60°, and 90°, the values are memorized and frequently pop up:

• For \( \sin 60^{\circ} \), the value is \( \frac{\sqrt{3}}{2} \), indicating the vertical component length in a unit circle.
• Conversely, \( \cos 60^{\circ} \) is \( \frac{1}{2} \), showcasing the horizontal component length.

From these, we use cofunction identities to find related values, such as:

• \( \sin 30^{\circ} = \cos 60^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

These specific values form the foundation for solving more intricate trigonometric problems.

Tangent and Cotangent

Tangent and cotangent are also crucial in trigonometric functions, focusing on the relationships between angles and triangle sides.

• The tangent of an angle is the ratio between the sine and cosine of that angle. Mathematically, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• With the given values \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) and \( \cos 60^{\circ} = \frac{1}{2} \), you find \( \tan 60^{\circ} = \frac{\sqrt{3}}{2} \div \frac{1}{2} = \sqrt{3} \).

The cotangent, on the other hand, is the reciprocal of the tangent:

• Given by \( \cot \theta = \frac{1}{\tan \theta} \).
• Thus, \( \cot 60^{\circ} = \frac{1}{\sqrt{3}} \), computed from the inverse of the tangent value.

Knowing these basic relationships aids in understanding more complex trigonometric concepts and their applications.