Exact Trigonometric Values for Special Angles

1. The Core Values: What You Need to Know

For the non-calculator paper, you must be able to instantly recall the sine, cosine, and tangent values for the key angles: 0°, 30°, 45°, 60°, and 90°. These values often involve fractions and surds (like $\sqrt{2}$ and $\sqrt{3}$).

Here is the table of values you need to master. Don't worry about memorizing it just yet; we'll cover several effective methods to learn and recall these.

Notice the patterns:

• The values for $\sin (\theta )$ from 0° to 90° are the same as the values for $\cos (\theta )$ from 90° to 0°.
• $\sin (30^{\circ })=\cos (60^{\circ })=\frac{1}{2}$.
• $\sin (60^{\circ })=\cos (30^{\circ })=\frac{\sqrt{3}}{2}$.
• $\sin (45^{\circ })=\cos (45^{\circ })$.

2. How to Remember the Values

Recalling these values quickly and accurately under exam pressure is crucial. Rote memorization can fail, so let's explore two more robust techniques. You can choose the one that works best for you.

Method 1: The Table Pattern

This method rebuilds the table using a simple counting pattern. It's systematic and easy to verify.

The core idea for sine is:

1. Write the numbers 0, 1, 2, 3, 4.
2. Square root them all: $\sqrt{0},\sqrt{1},\sqrt{2},\sqrt{3},\sqrt{4}$.
3. Divide them all by 2.
   This gives you $\sin (0^{\circ }),\sin (30^{\circ }),\sin (45^{\circ }),\sin (60^{\circ }),\sin (90^{\circ })$. The cosine values are the same, but in reverse order. The tangent values can be found using the identity $\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )}$.

Method 2: The Hand Trick

This is a fast, kinesthetic mnemonic using your left hand. It can be a lifesaver if you blank out during an exam.

To support this, here is a fantastic visual summary of the hand trick. You might want to practice using it a few times until it becomes second nature.

Choose whichever method feels more natural and reliable for you. The goal is to be able to find any value within seconds.

3. The 'Why': Deriving the Values from First Principles

Since you have a strong interest in maths and physics, just memorizing the values might not be fully satisfying. Understanding why $\sin (60^{\circ })$ is $\frac{\sqrt{3}}{2}$ deepens your knowledge and connects to the logical proofs you've mentioned you struggle with.

The values for 30°, 45°, and 60° can be derived from two "special" triangles.

1. The Isosceles Right-Angled Triangle:

   • Imagine a right-angled triangle with two equal sides of length 1.
   • The two non-right angles must both be 45°.
   • By Pythagoras' theorem, the hypotenuse is $\sqrt{1^2+1^2}=\sqrt{2}$.
   • Using SOHCAHTOA on this triangle, you can find that $\sin (45^{\circ })=\frac{O}{H}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

2. The Equilateral Triangle:

   • Imagine an equilateral triangle with side lengths of 2. All angles are 60°.
   • Slice it in half from top to bottom. This creates a right-angled triangle.
   • The angles are 30°, 60°, and 90°.
   • The hypotenuse is 2, the base is 1, and by Pythagoras, the height is $\sqrt{2^2-1^2}=\sqrt{3}$.
   • Using SOHCAHTOA on this triangle, you can find all the values for 30° and 60°. For example, $\cos (60^{\circ })=\frac{A}{H}=\frac{1}{2}$.

4. Application: Using Exact Values in Exam Questions

Now, let's put this knowledge into practice. In an exam, you'll see these values used in two main ways: direct calculations and solving triangles.

Type 1: Direct Calculation

You'll be asked to evaluate an expression. The method is simple: substitute the exact values and simplify.

Worked Example:
Without using a calculator, find the value of $2\sin (60^{\circ })-\tan (45^{\circ })$.

1. Recall the values:

   • $\sin (60^{\circ })=\frac{\sqrt{3}}{2}$
   • $\tan (45^{\circ })=1$

2. Substitute:
   $2\times \left(\frac{\sqrt{3}}{2}\right)-1$

3. Simplify:
   $\sqrt{3}-1$
   This is the final, exact answer.

Type 2: Solving Triangles

This is just like the SOHCAHTOA problems you're used to, but you substitute the exact value instead of using a calculator.

5. Practice Time

This topic is all about practice and recall. The more you use these values, the faster you'll become.

After this lesson, I highly recommend working through this sheet to test your speed and accuracy.

Conclusion

You have now learned not only what the exact trigonometric values are, but also how to remember them, where they come from, and how to apply them in exam-style questions. This is a key skill that turns a tricky non-calculator question into a straightforward process.

Key Takeaways:

• The exact values for sin, cos, and tan of 0°, 30°, 45°, 60°, and 90° are essential for the non-calculator paper.
• You can derive these values from an isosceles right-angled triangle and an equilateral triangle.
• You can quickly recall them using the table pattern or the hand trick.
• Application involves either substituting the values into expressions or using them in SOHCAHTOA problems to find exact lengths and angles.

Preview of the Next Lesson:
In our next lesson, we will move on to solving trigonometric equations for a given range of angles (e.g., find all values of $x$ between 0° and 360° for which $\cos (x)=0.5$). This topic will directly build on your new knowledge of exact values, which will help you find the initial solutions.