Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 60^{\circ} & & \frac{1}{2} & \sqrt{3} & & 2 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), \(\cot 60^{\circ} = \frac{\sqrt{3}}{3}\), \(\csc 60^{\circ} = \frac{2\sqrt{3}}{3}\).

Step 1: Identify \\(\sin 60^{\circ}\\)

Recall that \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). This is a standard trigonometric value for angles in a 30-60-90 triangle.

Step 2: Calculate \\(\tan 60^{\circ}\\)

The formula for the tangent is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). For \( 60^{\circ} \), substitute \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) and \( \cos 60^{\circ} = \frac{1}{2} \): \[\tan 60^{\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\] Thus, \( \tan 60^{\circ} = \sqrt{3} \). This value is already provided correctly in the table.

Step 3: Determine \\(\cot 60^{\circ}\\)

The cotangent is the reciprocal of the tangent: \( \cot 60^{\circ} = \frac{1}{\tan 60^{\circ}} \). Since \( \tan 60^{\circ} = \sqrt{3} \), we have: \[\cot 60^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\] Thus, \( \cot 60^{\circ} = \frac{\sqrt{3}}{3} \).

Step 4: Compute \\(\csc 60^{\circ}\\)

The cosecant is the reciprocal of the sine: \( \csc \theta = \frac{1}{\sin \theta} \). For \( 60^{\circ} \), substitute \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \): \[\csc 60^{\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\] Thus, \( \csc 60^{\circ} = \frac{2\sqrt{3}}{3} \).

Step 5: Verify \\(\sec 60^{\circ}\\)

The secant is the reciprocal of the cosine: \( \sec \theta = \frac{1}{\cos \theta} \). For \( 60^{\circ} \), \( \cos 60^{\circ} = \frac{1}{2} \) is given, so: \[\sec 60^{\circ} = \frac{1}{\frac{1}{2}} = 2\] This matches the value given in the table.

Key Concepts

Exact Values

Trigonometric functions often have special exact values, especially for commonly used angles like 30°, 45°, and 60°. These exact values can be remembered and don't require a calculator.

• \( ext{For } 60^{\circ}\), \(\sin 60^{\circ}\) is \(\frac{\sqrt{3}}{2}\). This comes from the properties of a 30-60-90 triangle, where the longer leg is \(\sqrt{3}\) times the shorter leg when the hypotenuse is 2.
• \(\cos 60^{\circ}\) is \(\frac{1}{2}\), aligning with the 30-60-90 triangle relationships.
• \( an 60^{\circ}\) equals \(\sqrt{3}\), which is the ratio of \(\sin\) over \(\cos\). It indicates how steep the line is at that angle.

Exact values help solve many math problems quickly if you know these values by heart. Memorizing them can simplify many algebra and geometry exercises.

Angle Measures

In the context of trigonometry, angles may be measured in degrees or radians. For practical reasons and initial learning, students often use degrees.

• Degree System: A full circle is 360°, making 60° a sixth of a circle. This makes it easy to divide and understand various geometric shapes.
• Radians System: Here, a full circle is \(2\pi\) radians. Hence, 60° converts to \(\pi/3\) radians (since \(60/360 \times 2\pi = \pi/3\)). Radians are used more in higher mathematics and physics due to their natural properties in calculus.

The angle measure affects trigonometric calculations, as certain functions and graphs use radians by default. However, tools and text allow for conversions.

Reciprocal Trigonometric Functions

Beyond the primary trigonometric functions (sine, cosine, tangent), there are reciprocal functions. These include cosecant, secant, and cotangent, each being the reciprocal of one of the primary functions.

• Cosecant (\(\csc\)): Derives from the sine function, \(\csc \theta = \frac{1}{\sin \theta}\). For \(60^{\circ}\), this value is \(\frac{2\sqrt{3}}{3}\).
• Secant (\(\sec\)): The reciprocal of cosine, expressed as \(\sec \theta = \frac{1}{\cos \theta}\). At \(60^{\circ}\), it becomes 2.
• Cotangent (\(\cot\)): Derived from the tangent, \(\cot \theta = \frac{1}{\tan \theta}\). For 60°, \(\cot\) is \(\frac{\sqrt{3}}{3}\).

Understanding these reciprocal functions broadens the range of equations and problems that can be solved, particularly in complex trigonomic and calculus applications.