Exact values of trigonometric functions

Welcome to the World of Exact Trig Values!

Hi there! Have you ever noticed that when you type sin(60) into your calculator, you get a long, messy decimal like 0.866025...? In Mathematics, we often prefer things to be neat and tidy. That is where Exact Values come in!

In this chapter, we are going to learn how to find the precise square-root versions of these numbers. Why? Because using \( \frac{\sqrt{3}}{2} \) is much more accurate than rounding a decimal. It’s like the difference between saying someone is "about 6 feet tall" and knowing their height is exactly 182.88 cm. Precision matters, especially in fields like engineering and architecture!

1. The "Big Three" Angles

For your OCR AS Level exam, you need to know the exact values for \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ, \) and \( 180^\circ \). These are the "VIPs" of the trigonometry world.

Don't worry if this seems like a lot to memorize! We can break them down using two simple "Special Triangles."

The 45° Isosceles Triangle

Imagine a right-angled triangle where the two shorter sides are both 1 unit long. Because the sides are equal, the angles must both be \( 45^\circ \).

Using Pythagoras' Theorem (\( a^2 + b^2 = c^2 \)):
\( 1^2 + 1^2 = 2 \)
So, the long side (hypotenuse) is \( \sqrt{2} \).

From this triangle, we get:
sin(45°) = \( \frac{Opposite}{Hypotenuse} \) = \( \frac{1}{\sqrt{2}} \) (or \( \frac{\sqrt{2}}{2} \))
cos(45°) = \( \frac{Adjacent}{Hypotenuse} \) = \( \frac{1}{\sqrt{2}} \) (or \( \frac{\sqrt{2}}{2} \))
tan(45°) = \( \frac{Opposite}{Adjacent} \) = \( \frac{1}{1} \) = 1

The 30° and 60° Triangle

Now, imagine an equilateral triangle where every side is 2 units long. All angles are \( 60^\circ \). If we chop it exactly down the middle, we get a right-angled triangle with angles of \( 30^\circ \) and \( 60^\circ \).

The base is now 1 unit (half of 2), and the hypotenuse is still 2 units. Using Pythagoras, the vertical height is \( \sqrt{3} \).

From this triangle, we get our 60° values:
sin(60°) = \( \frac{\sqrt{3}}{2} \)
cos(60°) = \( \frac{1}{2} \)
tan(60°) = \( \sqrt{3} \)

And our 30° values:
sin(30°) = \( \frac{1}{2} \)
cos(30°) = \( \frac{\sqrt{3}}{2} \)
tan(30°) = \( \frac{1}{\sqrt{3}} \) (or \( \frac{\sqrt{3}}{3} \))

Key Takeaway:

If you forget the values, just sketch the triangles! One is an isosceles triangle (1, 1, \( \sqrt{2} \)) and the other is half an equilateral triangle (1, \( \sqrt{3} \), 2).

2. The Easy Memory Trick: "The Square Root Count"

If triangles aren't your thing, try this simple pattern for Sine and Cosine. It’s like counting to four!

For Sine, as the angle goes \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \), the values are always \( \frac{\sqrt{n}}{2} \):

1. sin(0°) = \( \frac{\sqrt{0}}{2} \) = 0
2. sin(30°) = \( \frac{\sqrt{1}}{2} \) = \( \frac{1}{2} \)
3. sin(45°) = \( \frac{\sqrt{2}}{2} \)
4. sin(60°) = \( \frac{\sqrt{3}}{2} \)
5. sin(90°) = \( \frac{\sqrt{4}}{2} \) = \( \frac{2}{2} \) = 1

Quick Tip: For Cosine, just do the same thing but in reverse order! Cos(0°) is 1, and Cos(90°) is 0.

3. Summary Table of Exact Values

Here is a quick reference for the values you need to know. Tip: Write this out a few times to help it stick!

Angle 0°: sin = 0 | cos = 1 | tan = 0
Angle 30°: sin = \( \frac{1}{2} \) | cos = \( \frac{\sqrt{3}}{2} \) | tan = \( \frac{1}{\sqrt{3}} \)
Angle 45°: sin = \( \frac{\sqrt{2}}{2} \) | cos = \( \frac{\sqrt{2}}{2} \) | tan = 1
Angle 60°: sin = \( \frac{\sqrt{3}}{2} \) | cos = \( \frac{1}{2} \) | tan = \( \sqrt{3} \)
Angle 90°: sin = 1 | cos = 0 | tan = undefined
Angle 180°: sin = 0 | cos = -1 | tan = 0

Did you know? Tan(90°) is undefined because \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). At 90°, you would be dividing by zero (\( \frac{1}{0} \)), which is a mathematical "no-no"!

4. Working with Multiples

The syllabus also requires you to know values for multiples of these angles (like 120°, 135°, or 210°). You don't need a new table for these! You just need to use the symmetry of the trig graphs or a unit circle.

Step 1: Find the "Reference Angle" (how far the angle is from the horizontal x-axis).
Step 2: Use your exact value table for that reference angle.
Step 3: Decide if the value should be positive or negative based on which "quadrant" it is in (remember the CAST diagram).

Example: Find the exact value of sin(120°).
1. 120° is in the second quadrant. The distance to the x-axis (180°) is 60°.
2. We know sin(60°) is \( \frac{\sqrt{3}}{2} \).
3. In the second quadrant, Sine is positive. So, sin(120°) = \( \frac{\sqrt{3}}{2} \).

5. Common Mistakes to Avoid

1. Mixing up Sin and Cos: Remember that as the angle gets bigger (from 0 to 90), Sin starts at 0 and goes up to 1, while Cos starts at 1 and goes down to 0.
2. Forgetting the \( \sqrt{} \) symbol: Students often write 3/2 instead of \( \sqrt{3}/2 \). Always double-check your roots!
3. Calculator mode: Ensure your calculator is in Degrees mode, not Radians, when working with these specific angles.

Quick Review:

1. Exact values use surds (roots) and fractions to maintain 100% accuracy.
2. Use Special Triangles to derive values if you forget them.
3. The CAST diagram helps you find exact values for angles larger than 90°.
4. Practice makes perfect! The more you use these values in equations, the more they will feel like second nature.