What is the Derivation of Trigonometric Ratios for 45 Degrees?

S6-SA2-0102

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The derivation of trigonometric ratios for 45 degrees involves finding the values of sin 45, cos 45, and tan 45 by using a special right-angled triangle. We use an isosceles right-angled triangle, where two sides are equal and one angle is 90 degrees, making the other two angles 45 degrees each.

Simple Example

Quick Example

Imagine you have a square-shaped 'rangoli' design. If you draw a line diagonally across the square, you divide it into two identical right-angled triangles. Each of these triangles will have angles of 45, 45, and 90 degrees, which is the exact shape we use to find these ratios!

Worked Example

Step-by-Step

Let's derive the trigonometric ratios for 45 degrees.

Step 1: Consider an isosceles right-angled triangle ABC, where angle B is 90 degrees, and angle A = angle C = 45 degrees.

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Step 2: Since it's an isosceles triangle, the sides opposite the equal angles are equal. So, let AB = BC = 'a' units.

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Step 3: Use the Pythagorean theorem (Hypotenuse^2 = Base^2 + Perpendicular^2) to find the hypotenuse AC. AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2.

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Step 4: So, AC = sqrt(2a^2) = a * sqrt(2) units.

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Step 5: Now, let's find the ratios for angle C (45 degrees). Opposite side = AB = a, Adjacent side = BC = a, Hypotenuse = AC = a * sqrt(2).

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Step 6: sin 45 = Opposite / Hypotenuse = a / (a * sqrt(2)) = 1 / sqrt(2).

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Step 7: cos 45 = Adjacent / Hypotenuse = a / (a * sqrt(2)) = 1 / sqrt(2).

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Step 8: tan 45 = Opposite / Adjacent = a / a = 1.

Answer: The trigonometric ratios for 45 degrees are sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2), and tan 45 = 1.

Why It Matters

These 45-degree ratios are crucial for engineers designing buildings or bridges, as they help calculate slopes and forces. In gaming and animation, knowing these angles helps characters move realistically. Even in satellite navigation (like your phone's GPS), these fundamental ratios are used in complex calculations to pinpoint locations.

Common Mistakes

MISTAKE: Assuming the hypotenuse is 'a' when the other two sides are 'a'. | CORRECTION: Remember the hypotenuse is always the longest side in a right-angled triangle and must be calculated using Pythagoras theorem (a * sqrt(2) in this case).

MISTAKE: Mixing up Opposite and Adjacent sides when calculating ratios. | CORRECTION: Always identify the angle first. The side directly across from it is 'Opposite', and the side next to it (not the hypotenuse) is 'Adjacent'.

MISTAKE: Forgetting to rationalize the denominator (e.g., leaving 1/sqrt(2) as is). | CORRECTION: While 1/sqrt(2) is mathematically correct, it's good practice to rationalize it to sqrt(2)/2 by multiplying numerator and denominator by sqrt(2).

Practice Questions

Try It Yourself

QUESTION: If the equal sides of an isosceles right-angled triangle are 5 cm each, what is the length of the hypotenuse? | ANSWER: 5 * sqrt(2) cm

QUESTION: In an isosceles right-angled triangle, if the hypotenuse is 10 units, what is the length of each of the other two equal sides? (Hint: Use Pythagoras theorem in reverse) | ANSWER: 10 / sqrt(2) = 5 * sqrt(2) units

QUESTION: A ladder leans against a wall, making a 45-degree angle with the ground. If the base of the ladder is 3 meters away from the wall, how long is the ladder? | ANSWER: 3 * sqrt(2) meters

MCQ

Quick Quiz

What is the value of sin 45 degrees?

sqrt(2)

1 / sqrt(2)

1 / 2

The Correct Answer Is:

For a 45-degree angle, the opposite side and adjacent side are equal. Using the Pythagorean theorem, the hypotenuse is sqrt(2) times one of these sides. So, sin 45 = Opposite/Hypotenuse = 1/sqrt(2).

Real World Connection

In the Real World

When a drone camera (like those used for wedding photography or mapping in India) needs to capture a perfect 45-degree angled shot, its flight path and camera tilt are calculated using these exact trigonometric ratios. This ensures the drone flies at the correct height and distance to get the desired view.

Key Vocabulary

Key Terms

ISOSCELES TRIANGLE: A triangle with two equal sides and two equal angles | HYPOTENUSE: The longest side of a right-angled triangle, opposite the 90-degree angle | PYTHAGOREAN THEOREM: In a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse | RATIONALIZE: To remove a radical (like sqrt) from the denominator of a fraction | TRIGONOMETRIC RATIOS: Ratios of sides of a right-angled triangle related to its angles (sin, cos, tan)

What's Next

What to Learn Next

Next, you should learn about the derivation of trigonometric ratios for 30 and 60 degrees. This will complete your understanding of special angles and help you solve even more complex problems in trigonometry!