What is the Value of tan 45 degrees?

S6-SA2-0294

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 concept in trigonometry. It represents the ratio of the side opposite to the angle to the side adjacent to the angle in a right-angled triangle, specifically when one of the acute angles is 45 degrees. Its value is always 1.

Simple Example

Quick Example

Imagine you are flying a kite, and the string makes an angle of 45 degrees with the ground. If the kite is directly above a point that is 10 meters away from you on the ground, then the height of the kite from the ground would also be 10 meters. This is because tan 45 degrees = height / distance on ground, and since tan 45 degrees is 1, height = distance on ground.

Worked Example

Step-by-Step

Let's find the value of tan 45 degrees using a right-angled isosceles triangle.
---Step 1: Draw a right-angled triangle ABC, with the right angle at B (90 degrees).
---Step 2: Since it's an isosceles right-angled triangle, the other two angles must be equal. So, Angle A = Angle C = (180 - 90) / 2 = 45 degrees.
---Step 3: In an isosceles right-angled triangle, the two sides forming the right angle are equal. Let AB = BC = 'x' units.
---Step 4: Recall the definition of tangent: tan(angle) = Opposite side / Adjacent side.
---Step 5: For Angle C (which is 45 degrees), the Opposite side is AB and the Adjacent side is BC.
---Step 6: So, tan 45 degrees = AB / BC = x / x.
---Step 7: When you divide a number by itself (and it's not zero), the result is 1.
---Answer: Therefore, the value of tan 45 degrees is 1.

Why It Matters

Understanding tan 45 degrees is crucial for solving many real-world problems in engineering and physics, especially those involving angles and distances. Engineers use it to design structures, and physicists apply it in calculating trajectories or forces, leading to careers in civil engineering, robotics, or even space technology.

Common Mistakes

MISTAKE: Confusing tan 45 with sin 45 or cos 45. | CORRECTION: Remember that tan 45 is 1, while sin 45 is 1/sqrt(2) and cos 45 is 1/sqrt(2). Each trigonometric ratio has a unique value for a given angle.

MISTAKE: Forgetting that tan is Opposite/Adjacent. | CORRECTION: Always recall the SOH CAH TOA mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

MISTAKE: Assuming tan 45 degrees is 0.5 because 45 is half of 90. | CORRECTION: Trigonometric values are not directly proportional to the angle measure. Always use the definitions or known values from the trigonometric table.

Practice Questions

Try It Yourself

QUESTION: If in a right-angled triangle, the side opposite to 45 degrees is 5 cm, what is the length of the adjacent side? | ANSWER: 5 cm

QUESTION: A ladder leans against a wall, making an angle of 45 degrees with the ground. If the base of the ladder is 3 meters from the wall, what is the height the ladder reaches on the wall? | ANSWER: 3 meters

QUESTION: In a right-angled triangle PQR, right-angled at Q, if angle P is 45 degrees and QR = 7 units, find the length of PQ. | ANSWER: PQ = 7 units