Trigonometric Ratios of Specific Angles

🔄 Quick Recap

In the previous section, we learned about the six trigonometric ratios (sin, cos, tan, cosec, sec, and cot) and how they relate the sides of a right triangle to its angles. Now, we'll find the exact values of these ratios for some commonly used angles.

📚 Why Learn Specific Angle Values?

Knowing the exact trigonometric values for common angles like 0°, 30°, 45°, 60°, and 90° is extremely useful because:

• You can solve many problems without a calculator
• These values appear frequently in mathematics, physics, and engineering
• They help you develop intuition about how trigonometric functions behave
• They form building blocks for understanding more complex trigonometric concepts

Understanding the Angle Chart

In the diagram above, you can see:

• The five standard angles (0°, 30°, 45°, 60°, 90°) represented visually in circles
• Each red line shows the angle measured from the positive x-axis
• Below the angles, you'll find a table with the values of sin, cos, and tan for each angle

Look at how the angles progress from left to right:

• At 0°, the line points directly to the right
• At 30°, the line tilts slightly upward
• At 45°, the line makes an equal angle with both axes
• At 60°, the line tilts more steeply upward
• At 90°, the line points directly upward

The table at the bottom shows the exact values. Notice the patterns:

• sin increases from 0 at 0° to 1 at 90°
• cos decreases from 1 at 0° to 0 at 90°
• tan increases from 0 at 0° and becomes undefined at 90°

Let's learn how to derive these values!

📚 Finding Trigonometric Ratios for 45°

Let's start with a 45° angle. We'll create a right triangle with one angle equal to 45°.

Since the angles in a triangle add up to 180°, and we already have a 90° angle (the right angle) and a 45° angle, the third angle must also be 45°.

In this special case, the two non-right angles are equal, which means the two sides opposite to these angles must also be equal (this is an isosceles right triangle).

Let's say both of these sides have length 'a'. We can use the Pythagorean theorem to find the hypotenuse:

Hypotenuse² = a² + a²
Hypotenuse² = 2a²
Hypotenuse = a√2

Now, we can find the trigonometric ratios:

sin 45° = Opposite/Hypotenuse = a/(a√2) = 1/√2 = √2/2

cos 45° = Adjacent/Hypotenuse = a/(a√2) = 1/√2 = √2/2

tan 45° = Opposite/Adjacent = a/a = 1

Notice something interesting? For a 45° angle, sin and cos have the same value (√2/2). This makes sense because in a 45-45-90 triangle, the two legs are equal, making the opposite and adjacent sides the same length.

The other three ratios are the reciprocals of these:

cosec 45° = 1/sin 45° = √2

sec 45° = 1/cos 45° = √2

cot 45° = 1/tan 45° = 1

📚 Finding Trigonometric Ratios for 30° and 60°

To find these values, we'll use an equilateral triangle, where all angles are 60° and all sides are equal.

Imagine an equilateral triangle with sides of length 2a. If we draw a line from one vertex straight down to the middle of the opposite side, we create two right triangles. This line is called the height or altitude of the triangle.

Because of the symmetry of an equilateral triangle:

• The height divides the base into two equal parts, each of length a
• The height creates two identical right triangles
• Each right triangle has angles of 30°, 60°, and 90°

We can use the Pythagorean theorem to find the height:

Height² = Hypotenuse² - Base²
Height² = (2a)² - a²
Height² = 4a² - a²
Height² = 3a²
Height = a√3

Now, in one of the right triangles:

• The hypotenuse is 2a
• One side (half the base) is a
• The other side (height) is a√3

For the 30° angle:

sin 30° = Opposite/Hypotenuse = a/(2a) = 1/2

cos 30° = Adjacent/Hypotenuse = (a√3)/(2a) = √3/2

tan 30° = Opposite/Adjacent = a/(a√3) = 1/√3 = √3/3

For the 60° angle in the same triangle:

sin 60° = Opposite/Hypotenuse = (a√3)/(2a) = √3/2

cos 60° = Adjacent/Hypotenuse = a/(2a) = 1/2

tan 60° = Opposite/Adjacent = (a√3)/a = √3

Notice an interesting pattern: sin 30° = cos 60° and cos 30° = sin 60°. This relationship always holds for complementary angles (angles that add up to 90°).

📚 Trigonometric Ratios for 0° and 90°

For these angles, we need to think about what happens to a right triangle as one of its angles approaches 0° or 90°.

For 0°:

Imagine a right triangle where one angle gets smaller and smaller, approaching 0°:

• The opposite side becomes very small, approaching 0
• The adjacent side gets closer to being the same length as the hypotenuse

This gives us:

sin 0° = 0
cos 0° = 1
tan 0° = 0
cosec 0° = Not defined (because 1/0 is undefined)
sec 0° = 1
cot 0° = Not defined (because 0/0 is undefined)

For 90°:

Now imagine a right triangle where one angle gets larger and larger, approaching 90°:

• The adjacent side becomes very small, approaching 0
• The opposite side gets closer to being the same length as the hypotenuse

This gives us:

sin 90° = 1
cos 90° = 0
tan 90° = Not defined (because 1/0 is undefined)
cosec 90° = 1
sec 90° = Not defined (because 1/0 is undefined)
cot 90° = 0

🖼️ Visual Aids: The Unit Circle

Another way to understand trigonometric ratios is through the unit circle - a circle with radius 1 centered at the origin of a coordinate system.

Let's understand what this diagram shows:

• The circle has its center at the origin (0,0) and has a radius of 1 unit (that's why it's called the "unit circle")
• The x and y axes divide the plane into four quadrants
• The green line from the origin to the point marked 45° shows an angle of 45 degrees from the positive x-axis
• The point where this line intersects the circle has coordinates (cos 45°, sin 45°)
• The orange dashed lines show how these coordinates are measured

For any angle θ on the unit circle:

• The x-coordinate of the point is cos θ
• The y-coordinate of the point is sin θ
• This means if you know the angle, you can find the exact point on the circle, and vice versa

For example, at 45 degrees:

• The coordinates are (cos 45°, sin 45°) = (√2/2, √2/2)
• Notice that the green line makes equal angles with both axes, so the point is equally distant from both axes

The unit circle provides a powerful way to visualize how trigonometric values change as angles change:

• At 0°, the point is at (1,0), so cos 0° = 1 and sin 0° = 0
• As the angle increases, the x-coordinate (cos) decreases and the y-coordinate (sin) increases
• At 90°, the point is at (0,1), so cos 90° = 0 and sin 90° = 1

⚖️ Quick Comparison/Summary Table

Here's a table summarizing all the trigonometric ratios for these special angles:

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	Not defined	1	Not defined
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	Not defined	1	Not defined	0

✅ Solved Examples

Example 1:

In a right triangle ABC, right-angled at B, if angle A = 60°, and AB = 8 cm, find the lengths of BC and AC.

Solution: Let's identify what we know:

• Angle A = 60°
• AB = 8 cm (the adjacent side to angle A)
• We need to find BC (opposite side) and AC (hypotenuse)

To find BC (opposite side to angle A), we can use the tangent ratio:

tan 60° = BC/AB
√3 = BC/8
BC = 8√3 cm ≈ 13.86 cm

To find AC (hypotenuse), we can use the cosine ratio:

cos 60° = AB/AC
1/2 = 8/AC
AC = 8 × 2 = 16 cm

We can verify our answer using the Pythagorean theorem:

AB² + BC² = AC²
8² + (8√3)² = 16²
64 + 192 = 256
256 = 256 ✓

Example 2:

Find the value of sin 60° × cos 30° + sin 30° × cos 60°.

Solution: Let's substitute the known values:

sin 60° × cos 30° + sin 30° × cos 60°
= (√3/2) × (√3/2) + (1/2) × (1/2)
= 3/4 + 1/4
= 1

This is an interesting result! In fact, for any angle A, sin A × cos (90°-A) + sin (90°-A) × cos A = 1.

🧪 Activity Time!

Make Your Own Angle Measurer (Clinometer)

Materials needed:

• A piece of cardboard
• A protractor
• A string (about 20 cm long)
• A small weight (like a washer or paper clip)
• Tape
• Marker or pen

Steps:

1. Trace a protractor onto the cardboard and cut it out
2. Mark the angles (0°, 30°, 45°, 60°, 90°) clearly on your cardboard protractor
3. Punch a small hole at the center point of the protractor's base
4. Tie one end of the string through this hole
5. Attach the weight to the other end of the string
6. Your clinometer is ready!

To use it:

1. Hold the clinometer at eye level and look through the straight edge at the top of the object you want to measure (like a tree or building)
2. The string with the weight will hang straight down due to gravity
3. Note the angle where the string crosses the protractor's scale
4. This angle is the angle of elevation to the top of the object
5. Measure the distance from where you're standing to the base of the object
6. Use the formula: height = distance × tan(angle) to calculate the height!

This simple tool lets you apply trigonometry to real-world measurements!

🧠 Memory Tricks

To remember the values of sin and cos for 0°, 30°, 45°, 60°, and 90°:

For sine values (0° to 90°):

• Think of the fractions: 0/2, 1/2, √2/2, √3/2, 2/2
• Notice the pattern? The numerator follows the sequence: 0, 1, √2, √3, 2
• The denominator is always 2

For cosine values (0° to 90°):

• These are the same as sine values but in reverse order
• That's because cos A = sin(90° - A)

For tangent values:

• tan 0° = 0
• tan 30° = 1/√3
• tan 45° = 1
• tan 60° = √3
• tan 90° = undefined

A helpful phrase to remember these special angles and their sine values: "0 hands, 1 hand, 2 hands, 3 hands, all hands" This helps recall sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1

🤔 Think About It!

1. How do the values of sine and cosine change as the angle increases from 0° to 90°? Can you sketch a rough graph?

2. Why do you think tan 90° is undefined? What happens geometrically as an angle approaches 90°?

3. Can you find a pattern in the values of sin θ and cos (90° - θ)? Why do you think this relationship exists?

4. In the unit circle, what would happen if we extended angles beyond 90°? How would the trigonometric values behave?

🔜 What Next?

In the next section, we'll learn about trigonometric identities - special relationships between trigonometric functions that are always true. These identities will help us simplify complex expressions and solve more advanced problems. Understanding these identities is like having powerful shortcuts in mathematics that make solving problems much easier!