What is the Value of cos 45?

S6-SA2-0045

Grade Level:

Class 10

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

Definition

What is it?

The value of cos 45 degrees is a specific ratio in a right-angled triangle. It represents the ratio of the length of the adjacent side to the length of the hypotenuse when one of the acute angles is 45 degrees. This value is a fixed constant, always 1/sqrt(2) or approximately 0.707.

Simple Example

Quick Example

Imagine you're flying a kite on a breezy day. If the kite string makes an angle of 45 degrees with the ground, and you want to know how far horizontally the kite is from you compared to the length of the string, that ratio is cos 45. If the string is 100 meters long, the horizontal distance would be 100 * cos 45 meters.

Worked Example

Step-by-Step

Let's find the value of cos 45 degrees using a right-angled isosceles triangle.

Step 1: Draw a right-angled triangle ABC, with the right angle at B.
---Step 2: Since it's an isosceles right-angled triangle, two angles must be equal. Let angle A = angle C = 45 degrees.
---Step 3: In an isosceles right-angled triangle, the sides opposite the equal angles are also equal. So, let AB = BC = 'a' units.
---Step 4: Use the Pythagorean theorem to find the hypotenuse AC. AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2. So, AC = sqrt(2a^2) = a * sqrt(2) units.
---Step 5: Recall the definition of cosine: cos(angle) = (Adjacent side) / (Hypotenuse).
---Step 6: For angle A (which is 45 degrees), the adjacent side is AB = 'a' and the hypotenuse is AC = a * sqrt(2).
---Step 7: Therefore, cos 45 degrees = AB / AC = a / (a * sqrt(2)).
---Step 8: Cancel out 'a' from the numerator and denominator. cos 45 degrees = 1 / sqrt(2).

Answer: The value of cos 45 degrees is 1/sqrt(2).

Why It Matters

Understanding cos 45 is crucial in many fields, from designing buildings to launching rockets. Engineers use trigonometry to calculate forces and distances in structures, and physicists use it to analyze wave patterns and trajectories. Knowing these values helps in careers like civil engineering, robotics, and even game development.

Common Mistakes

MISTAKE: Remembering sin 45 as 1/sqrt(2) and cos 45 as sqrt(3)/2 | CORRECTION: Both sin 45 and cos 45 are equal to 1/sqrt(2). It's a unique angle where sine and cosine values are the same.

MISTAKE: Writing cos 45 as sqrt(2) | CORRECTION: The value is 1 divided by sqrt(2), not just sqrt(2). Remember it's a ratio, usually less than 1.

MISTAKE: Forgetting to rationalize the denominator and leaving it as 1/sqrt(2) when asked for a simplified form | CORRECTION: While 1/sqrt(2) is correct, it's often preferred to rationalize it to sqrt(2)/2 by multiplying both numerator and denominator by sqrt(2).

Practice Questions

Try It Yourself

QUESTION: If the hypotenuse of a right-angled triangle is 10 cm and one angle is 45 degrees, what is the length of the adjacent side to that angle? | ANSWER: 10 * (1/sqrt(2)) cm or 5 * sqrt(2) cm

QUESTION: What is the exact value of cos 45 degrees multiplied by 2? | ANSWER: 2 * (1/sqrt(2)) = sqrt(2)

QUESTION: If sin 45 degrees = x, what is the value of x^2 + cos 45 degrees? | ANSWER: (1/sqrt(2))^2 + (1/sqrt(2)) = 1/2 + 1/sqrt(2) = (1 + sqrt(2))/2

MCQ

Quick Quiz

What is the rationalized value of cos 45 degrees?

1/sqrt(2)

sqrt(2)/2

sqrt(2)

The Correct Answer Is:

Option B, sqrt(2)/2, is the rationalized form of 1/sqrt(2). Options A is correct but not rationalized, and C and D are incorrect values for cos 45.

Real World Connection

In the Real World

Imagine an ISRO scientist calculating the trajectory of a satellite. If a component needs to be tilted at 45 degrees, they use cos 45 to figure out its horizontal projection. Similarly, a civil engineer designing a ramp or a bridge might use these trigonometric values to ensure stability and proper angles.

Key Vocabulary

Key Terms

COSINE: Ratio of adjacent side to hypotenuse in a right triangle | HYPOTENUSE: Longest side of a right triangle, opposite the right angle | ADJACENT SIDE: Side next to a given angle in a right triangle (not the hypotenuse) | RATIONALIZE: To remove a radical from the denominator of a fraction | TRIGONOMETRY: Branch of mathematics dealing with the relations of the sides and angles of triangles.

What's Next

What to Learn Next

Great job understanding cos 45! Next, you should explore the values of sin 30, cos 60, and tan 90 degrees. These special angles are frequently used together and will help you solve more complex trigonometry problems in Class 10 and beyond.