What are Trigonometric Ratios for 45 Degrees?

S6-SA2-0017

Grade Level:

Class 10

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

Definition

What is it?

Trigonometric ratios for 45 degrees are special values (like sin, cos, tan) that relate the sides of a right-angled triangle where one of the angles is exactly 45 degrees. These values are fixed and always the same for any 45-degree angle, making calculations easier.

Simple Example

Quick Example

Imagine you're flying a kite, and the string makes a 45-degree angle with the ground. If the kite is directly above a point 10 meters away from you on the ground, then because it's a 45-degree angle, the kite's height would also be 10 meters. This is because for 45 degrees, the opposite side and adjacent side are equal, making tan(45) = 1.

Worked Example

Step-by-Step

Let's find the trigonometric ratios for 45 degrees using a right-angled isosceles triangle.

Step 1: Consider a right-angled triangle ABC, with angle B = 90 degrees and angle A = 45 degrees.
---Step 2: Since the sum of angles in a triangle is 180 degrees, angle C = 180 - 90 - 45 = 45 degrees.
---Step 3: Because angle A = angle C = 45 degrees, the sides opposite to these angles must be equal. So, AB = BC. Let's assume AB = BC = 'a' units.
---Step 4: Now, we find the hypotenuse AC using the Pythagorean theorem: AC^2 = AB^2 + BC^2.
---Step 5: Substitute the values: AC^2 = a^2 + a^2 = 2a^2. So, AC = sqrt(2a^2) = a*sqrt(2).
---Step 6: Now we can find the ratios:
sin(45) = Opposite/Hypotenuse = BC/AC = a / (a*sqrt(2)) = 1/sqrt(2)
cos(45) = Adjacent/Hypotenuse = AB/AC = a / (a*sqrt(2)) = 1/sqrt(2)
tan(45) = Opposite/Adjacent = BC/AB = a / a = 1
---Step 7: The reciprocal ratios are:
cosec(45) = 1/sin(45) = sqrt(2)
sec(45) = 1/cos(45) = sqrt(2)
cot(45) = 1/tan(45) = 1

Answer: sin(45) = 1/sqrt(2), cos(45) = 1/sqrt(2), tan(45) = 1.

Why It Matters

Understanding these fixed ratios is crucial for engineers designing buildings, as they help calculate slopes and heights. In physics, these ratios are used to break down forces into components, which is vital for understanding how things move. Even in game development, these ratios help create realistic movements and angles for characters and objects.

Common Mistakes

MISTAKE: Confusing opposite and adjacent sides | CORRECTION: Always identify the angle you are working with first. The 'opposite' side is directly across from it, and the 'adjacent' side is next to it (but not the hypotenuse).

MISTAKE: Forgetting that sin(45) and cos(45) are equal | CORRECTION: Remember that in a 45-45-90 triangle, the two shorter sides are equal, which directly leads to sin(45) = cos(45) = 1/sqrt(2).

MISTAKE: Mixing up the reciprocal ratios (cosec, sec, cot) | CORRECTION: Remember that cosec is 1/sin, sec is 1/cos, and cot is 1/tan. If you know sin, cos, tan, just flip them to get the reciprocals.

Practice Questions

Try It Yourself

QUESTION: If the hypotenuse of a right-angled triangle with a 45-degree angle is 10 units, what is the length of the side opposite the 45-degree angle? | ANSWER: 5*sqrt(2) units (approx 7.07 units)

QUESTION: A ladder leans against a wall, making a 45-degree angle with the ground. If the base of the ladder is 6 meters away from the wall, what is the length of the ladder? | ANSWER: 6*sqrt(2) meters (approx 8.48 meters)

QUESTION: In a right-angled triangle PQR, angle Q is 90 degrees, and angle P is 45 degrees. If PQ = 8 cm, find the value of PR * sin(R). | ANSWER: 8 cm

MCQ

Quick Quiz

What is the value of tan(45 degrees)?

1/sqrt(2)

sqrt(2)

The Correct Answer Is:

For a 45-degree angle, the opposite side and the adjacent side in a right-angled triangle are equal. Since tan is Opposite/Adjacent, their ratio is 1.

Real World Connection

In the Real World

When ISRO scientists launch rockets, they use trigonometry to calculate the trajectory and angles needed for the rocket to reach orbit correctly. Surveyors also use these fixed ratios when measuring land or designing roads, especially when dealing with slopes that might be at a 45-degree angle.

Key Vocabulary

Key Terms

Right-angled triangle: A triangle with one angle exactly 90 degrees | Hypotenuse: The longest side of a right-angled triangle, opposite the 90-degree angle | Opposite side: The side directly across from a given angle | Adjacent side: The side next to a given angle, not the hypotenuse | Pythagorean theorem: A formula (a^2 + b^2 = c^2) relating the sides of a right-angled triangle

What's Next

What to Learn Next

Great job understanding 45-degree ratios! Next, you should learn about trigonometric ratios for 30 and 60 degrees. These are also special angles, and understanding them will complete your knowledge of basic trigonometric values for common angles.