Tan 45 Degrees: Value of tan 45 with Proof, Examples and FAQ

Tan 45 degrees

The value of the tangent of the angle 45° in the right-angled triangle is called tan of angle 45 degrees. The tangent of angle 45° is a value representing the ratio of the opposite side’s length to the adjacent side’s length with respect to the considered angle.

Value of Tan 45°

The exact value of tan (45°) is 1 in both fraction and decimal form.

tan (60°) = tan π/4 = 1

Proof

The exact value of tan π/4 can be derived using three methods explained below.

• Theoretical Method

We can derive the exact value of tan (45°) by using the property of the right-angled triangle. 

According to the right-angled triangle property, if an angle of the triangle is 45°, the triangle forms an isosceles right triangle (as shown in the figure)

Take, the length of adjacent and the opposite sides to the angle QPR as ‘l’ and the length of the hypotenuse to the angle QPR as ‘r’.

tan θ = (Length of opposite side) / (Length of the adjacent side)

Here,

tan (45°) = QR/PR

tan (45°) = l/l = 1

Therefore, tan (45°) =1

•  Practical Method

You can also find the value of the tangent of angle 45° practically by constructing a right-angled triangle with a 45° angle by geometrical tools.

Draw a straight horizontal line from Point H and then construct an angle of 45° using the protractor.

Set the compass to any length by a ruler. Here, the compass is set to 4.8 cm. Now, draw an arc on the 45° angle line from point H, and it intersects the line at the point I.

Finally, draw a perpendicular line on the horizontal line from the point I, and it intersects the horizontal line at point J perpendicularly. Thus, a right-angled triangle ∆HIJ is formed. 

Now,  calculate the value of the tangent of 45 degrees and for this, measure the length of the adjacent side (HJ) by a ruler. You will observe that the length of the opposite side (IJ) is 3.3 cm, and the length of the adjacent side is 3.37 cm in this example.

Now, find the ratio of lengths of the opposite side to the adjacent side and get the value of the tangent of angle 45°.

tan (45°) = IJ/GJ = (3.3)/(3.37)

So, tan (45°) = 0.97922… ≈ 1

• Trigonometric Method

We can prove the value of tan (45°) with a trigonometric approach.

we know that sin 45° = cos 45° = 1/√2

Also, by trigonometric identities,

sin x/cos x = tan x

Put x = 45°

tan (45°) = sin (45°)/cos (45°)

Put the values of sin 45° and cos 45°

tan (45°) = (1/√2)/(1/√2)

tan (45°) = 1

Hence, we proved the value of tan (45°) using different approaches.

Example

1. Evaluate: tan 45° + cos 60°

Solution:

We know that tan (45°) = 1 and cos (60°) = 1/2
So, tan (45°) + cos (60°)
= 1 + 1/2
= 3/2

2. Evaluate: 2 tan 45° – 2 sin 30°

Solution:

We know that tan (45°) = 1 and sin (30°) = 1/2
So, 2 tan (45°) – 2 sin (30°)
= 2 – 2(1/2)
= 2 – 1
= 1