What is the Value of tan 45?

S6-SA2-0046

Grade Level:

Class 10

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

Definition

What is it?

The value of tan 45 degrees is a fundamental trigonometric ratio. It represents the ratio of the side opposite to the 45-degree angle to the side adjacent to it in a right-angled triangle. Its exact value is 1.

Simple Example

Quick Example

Imagine you're flying a kite, and its string makes a 45-degree angle with the ground. If the height of the kite above the ground is exactly the same as the horizontal distance from you to the point directly below the kite, then the 'tan' of that angle is 1. It's like saying if the 'opposite' and 'adjacent' sides are equal, their ratio is 1.

Worked Example

Step-by-Step

Let's find the value of tan 45 degrees using a right-angled isosceles triangle.
1. Consider a right-angled triangle ABC, where angle B = 90 degrees.
2. Let angle A = 45 degrees. Since the sum of angles in a triangle is 180 degrees, angle C must also be 45 degrees (180 - 90 - 45 = 45).
3. Because angles A and C are equal (both 45 degrees), the sides opposite to them must also be equal. So, side BC (opposite to A) = side AB (opposite to C).
4. Let's assume BC = AB = 'a' units.
5. Recall that tan(angle) = Opposite side / Adjacent side.
6. For angle A (45 degrees), the opposite side is BC = 'a', and the adjacent side is AB = 'a'.
7. So, tan 45 degrees = BC / AB = a / a.
8. Therefore, tan 45 degrees = 1.
Answer: The value of tan 45 degrees is 1.

Why It Matters

Understanding tan 45 is crucial for solving many real-world problems in engineering and physics. For example, it helps engineers calculate slopes for roads or ramps, ensuring they are safe and stable. Architects use it to design roofs and structures, and even game developers use trigonometry for character movements and camera angles.

Common Mistakes

MISTAKE: Confusing tan 45 with sin 45 or cos 45, which have different values. | CORRECTION: Remember the SOH CAH TOA mnemonic: Tan is Opposite/Adjacent. Always refer to the specific trigonometric ratio.

MISTAKE: Forgetting that tan 45 is exactly 1, and trying to calculate it using a calculator for simple problems. | CORRECTION: Memorize the values of common angles like 0, 30, 45, 60, and 90 degrees for sin, cos, and tan. This saves time and prevents errors.

MISTAKE: Assuming tan 45 is 1 only in a specific type of triangle. | CORRECTION: The value of tan 45 degrees is always 1, regardless of the size of the right-angled triangle, as long as one of its acute angles is 45 degrees. The ratio of opposite to adjacent side will always be 1.

Practice Questions

Try It Yourself

QUESTION: If in a right-angled triangle, the angle is 45 degrees, and the opposite side is 7 cm, what is the length of the adjacent side? | ANSWER: 7 cm

QUESTION: A ladder leans against a wall, making a 45-degree angle with the ground. If the base of the ladder is 5 meters away from the wall, how high does the ladder reach on the wall? | ANSWER: 5 meters

QUESTION: In a right-angled triangle PQR, angle Q is 90 degrees. If angle P is 45 degrees, and the hypotenuse PR is 10*sqrt(2) units, find the length of PQ. (Hint: Use sin or cos 45 first to find PQ, then confirm with tan 45 relationship). | ANSWER: 10 units

MCQ

Quick Quiz

What is the value of tan 45 degrees?

sqrt(3)

1/sqrt(3)

The Correct Answer Is:

The value of tan 45 degrees is exactly 1. This comes from the ratio of the opposite side to the adjacent side in a right-angled triangle with a 45-degree angle, where these two sides are equal.

Real World Connection

In the Real World

Imagine an architect designing a ramp for a hospital in a Tier 2 city like Nashik. If the architect wants the ramp to have a 45-degree slope for easy access, they know that for every 1 meter the ramp goes up (vertical height), it must also go 1 meter horizontally. This 1:1 ratio is directly linked to tan 45 = 1, ensuring the ramp is not too steep or too flat.

Key Vocabulary

Key Terms

TRIGONOMETRY: A branch of mathematics dealing with the relationships between the sides and angles of triangles. | RATIO: A comparison of two numbers by division. | RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees. | OPPOSITE SIDE: The side across from a given angle in a right-angled triangle. | ADJACENT SIDE: The side next to a given angle in a right-angled triangle, not the hypotenuse.

What's Next

What to Learn Next

Great job understanding tan 45! Next, you should explore the values of tan for other special angles like 30 and 60 degrees. This will build your foundation in trigonometry and help you solve more complex problems involving heights, distances, and angles in your daily life and future studies.