8.3. Trigonometric Ratios of Some Specific Angles

In Δ ABC, right-angled at B, if one angle is 45°, then the other angle is also 45°, i.e., ∠ A = ∠ C = 45° (see Fig 8.14).

So, BC = AB (Why?)

Now, Suppose BC = AB = a.

Then by Pythagoras Theorem, AC² = AB² + BC² = a² + a² = 2a², and, therefore, AC = a√2.

Using the definitions of the trigonometric ratios, we have:

• sin 45° = BC/AC = a / (a√2) = 1/√2
• cos 45° = AB/AC = a / (a√2) = 1/√2
• tan 45° = BC/AB = a/a = 1
• cosec 45° = 1/sin 45° = √2,
• sec 45° = 1/cos 45° = √2,
• cot 45° = 1/tan 45° = 1.

In a right triangle where one angle is 45°, both other angles must also be 45° making this a special isosceles triangle. The lengths of the two legs (BC and AB) are equal. By using the Pythagorean theorem, we find the length of the hypotenuse (AC) as 'a√2', given that both legs are of equal length 'a'. The trigonometric ratios can then be calculated, revealing that sine and cosine of 45° are both equal to '1/√2', which simplifies to approximately 0.707.

Imagine you're climbing a staircase where each step makes a 45° angle with the ground. Since every time you go up one step (the height), you also move out one step (the horizontal distance), both distances are equal. If you used a ruler to measure these steps, both would be the same, reinforcing that the sine and cosine of those angles are equal.

Let us now calculate the trigonometric ratios of 30° and 60°. Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore, ∠ A = ∠ B = ∠ C = 60°.

Draw the perpendicular AD from A to the side BC (see Fig. 8.15).

Now, Δ ABD ≅ Δ ACD (Why?) Thus, BD = DC and ∠ BAD = ∠ CAD (CPCT).

Now observe that: Δ ABD is a right triangle, right-angled at D with ∠ BAD = 30° and ∠ ABD = 60°. As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle. So, let us suppose that AB = 2a.

Then, BD = BC= a/2 and AD² = AB² – BD² = (2a)² – (a)² = 3a², Therefore, AD = a√3. Now, we have:

• sin 30° = BD/AB = a / (2a) = 1/2,
• cos 30° = AD/AB = (a√3)/(2a) = √3/2,
• tan 30° = BD/AD = a/(a√3) = 1/√3.

In the triangle, we derive the lengths needed to calculate the trigonometric ratios for 30° and 60°. Since we form a right triangle from half of the equilateral triangle, we can use properties of the angles and the lengths of the sides to derive sine and cosine values specifically tied to these angles. The side relationships reveal that 30° has a sine of 1/2 and a cosine of √3/2, while the tangent is 1/√3, making them interconnected using the sides drawn from a common height.

Think of a tall tree. If you measure the height of the tree and find a point where the angle from your eyes to the top of the tree is 30°, you will be standing at half the tree's height (where the sine values apply). A triangle made from you to the top of the tree forms similar ratios that reflect these trigonometric values.

Let us see what happens to the trigonometric ratios of angle A, if it is made smaller and smaller in the right triangle ABC (see Fig. 8.16), till it becomes zero. As ∠ A gets smaller and smaller, the length of the side BC decreases. The point C gets closer to point B, and finally when ∠ A becomes very close to 0°, AC becomes almost the same as AB (see Fig. 8.17).

When ∠ A is very close to 0°, BC gets very close to 0 and so the value of sin A = BC/AC is very close to 0. Also, when ∠ A is very close to 0°, AC is nearly the same as AB and so the value of cos A = AC/AB is very close to 1. This helps us to see how we can define the values of sin A and cos A when A = 0°.

We define: sin 0° = 0 and cos 0° = 1. Using these, we have :

• tan 0° = sin 0° / cos 0° = 0,
• cot 0° is not defined,
• sec 0° = 1/cos 0° = 1,
• cosec 0° is not defined.

As angle A approaches 0°, we intuitively understand that the height from the base converges to a line, leading to near-zero sine while the cosine ratio remains close to 1. These limits help us differenciate between values of these trigonometric functions at boundaries. At 0°, we can establish known values: sin 0°=0 denotes the absence of height, while cos addresses the full horizontal stretch.

Picture diving from a diving board. As you lean forward to dive, you angle towards the water (θ approach 0°). Ultimately, when you dive straight, you're close to the horizontal which represents cos (like being flat) and your height drops to zero, portrayed as sin (falling into the water).

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