Special Angles

Special Angles are angles that act as integer multiples of \(\pi/6\) radians (or \(30^{\circ}\)) and \(\pi/4\) radians (or \(45^{\circ}\)). They're classified as "special" since they're simple to work with without use of a calculator and provide precise answers.

Outlined below is a table outlining the trigonometric values pertaining to their respective trigonometric functions and ratios:

	\(\boldsymbol{\sin}\)	\(\boldsymbol{\cos}\)	\(\boldsymbol{\tan}\)	\(\boldsymbol{\csc}\)	\(\boldsymbol{\sec}\)	\(\boldsymbol{\cot}\)
\(\boldsymbol{0 \; [\textbf{rads}]}\)	\(0\)	\(1\)	\(0\)	N/A	\(1\)	N/A
\(\boldsymbol{\cfrac{\pi}{6} \; [\textbf{rads}]}\)	\(\cfrac{1}{2}\)	\(\cfrac{\sqrt{3}}{2}\)	\(\cfrac{\sqrt{3}}{3}\)	\(2\)	\(\cfrac{2\sqrt{3}}{3}\)	\(\sqrt{3}\)
\(\boldsymbol{\cfrac{\pi}{4} \; [\textbf{rads}]}\)	\(\cfrac{\sqrt{2}}{2}\)	\(\cfrac{\sqrt{2}}{2}\)	\(1\)	\(\sqrt{2}\)	\(\sqrt{2}\)	\(1\)
\(\boldsymbol{\cfrac{\pi}{3} \; [\textbf{rads}]}\)	\(\cfrac{\sqrt{3}}{2}\)	\(\cfrac{1}{2}\)	\(\sqrt{3}\)	\(\cfrac{2\sqrt{3}}{3}\)	\(2\)	\(\cfrac{\sqrt{3}}{3}\)

Special Triangles

Using Right Triangles, we can easily identify how to identify the trigonometric ratios of \(30°\), \(45°\), and \(60°\):

It doesn't matter what the dimensions of the triangle are when measuring the angles since the ratios will always be the exact same. When evaluating a trigonometric value (i.e. \(\cos(30°)\)), make sure to display it as a trigonometric ratio (i.e. \(\cfrac{\sqrt{3}}{2}\)) rather than the exact amount to prevent rounding errors.

Example

Use the Special Angles chart and the Unit Circle to determine the exact values of the primary trigonometric ratios for \(\cfrac{4\pi}{3}\).

Using the Special Angles chart, we can determine that \(\cfrac{4π}{3}\) is a multiple of \(\cfrac{\pi}{3}\). Using the Unit Circle, we can also determine that this ratio lies in Quadrant \(3\).

Using this information, we can determine the exact values of these ratios:

\(\sin\left(\cfrac{4\pi}{3}\right) = \boldsymbol{-\cfrac{\sqrt{3}}{2}}\)

\(\cos\left(\cfrac{4\pi}{3}\right) = \boldsymbol{-\cfrac{\sqrt{1}}{2}}\)

\(\tan\left(\cfrac{4\pi}{3}\right) = \boldsymbol{-\sqrt{3}}\)

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Evaluating Trigonometric Expressions

In order to properly evaluate and solve trigonometric expressions, there are a few strategies that we can consider to help simplify the process:

1. Evaluate each respective trigonometric ratio first. This will provide you the values required to solve the rest of the expression
2. Use the Unit Circle and Special Angles as references. These will provide with the exact values and signs of the respective trigonometric ratios
3. Use BEDMAS. It’s important to remember to multiply and/or divide expressions before adding and/or subtracting them
4. Rationalize denominators when necessary. This will make the denominator easier to evaluate

Example

Determine an exact value for the expression \(\cfrac{\cos\left(\cfrac{\pi}{3}\right)}{\sin\left(\cfrac{\pi}{4}\right) \cdot \tan\left(\cfrac{\pi}{3}\right)}\).

First, we can evaluate each respective trigonometric ratio. For \(\cos(\pi/6)\), we can evaluate this ratio using its corresponding special triangle. We can use SOH CAH TOA to determine that the ratio is \(1/2\). Since the angle is located in Quadrant 1, we can use CAST to determine that \(\cos(\pi/6)\) is positive.

For \(\sin(\pi/4)\), we can evaluate this ratio using its corresponding special triangle. We can use SOH CAH TOA to determine that the ratio is \(\sqrt{2}/2\). Since the angle is located in Quadrant 1, we can use CAST to determine that \(\sin(\pi/4)\) is positive.

For \(\tan(\pi/3)\), we can evaluate this ratio using its corresponding special triangle. We can use SOH CAH TOA to determine that the ratio is \(\sqrt{3}\). Since the angle is located in Quadrant 1, we can use CAST to determine that \(\tan(\pi/3)\) is positive.

Based on what we determined above, we can rewrite the expression as such:

\(= \cfrac{\cfrac{1}{2}}{\cfrac{\sqrt{2}}{2} \cdot \sqrt{3}}\)

Next, we can simplify the expression by multiplying the denominator:

\(= \cfrac{\cfrac{1}{2}}{\cfrac{\sqrt{6}}{2}}\)

Then, we can multiply the numerator by the reciprocal of the denominator:

\(= \cfrac{1}{\cancel{2}} \cdot \cfrac{\cancel{2}}{\sqrt{6}}\)

\(= \cfrac{1}{\sqrt{6}}\)

Finally, we can rationalize the denominator by multiplying both the numerator and denominator by \(\sqrt{6}\):

\(= \cfrac{1}{\sqrt{6}} \cdot \cfrac{\sqrt{6}}{\sqrt{6}}\)

\(= \cfrac{6}{\sqrt{6}}\)

Therefore, we can determine that the exact value for\(\cfrac{\cos\left(\cfrac{\pi}{3}\right)}{\sin\left(\cfrac{\pi}{4}\right) \cdot \tan\left(\cfrac{\pi}{3}\right)}\) is \(\boldsymbol{\cfrac{6}{\sqrt{6}}}\).

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