# If tan A = 4/3, then find the value of cos A

Trigonometry is an important branch of mathematics, it deals with the relationship of sides with angles in a right-angle triangle. Trigonometry is important in physics also, it is used to find, the height of towers, the distance between stars, or in the Navigation systems.** Trigonometry is predicated on the principle that “If two triangles have an equivalent set of angles then their sides are within the same ratio”**. Side lengths are often different but side ratios are equivalent.

### Trigonometric Functions

Trigonometric functions or circular functions or trigonometric ratios show **the relationship of angles and sides**. These trigonometric ratios are obtained by taking ratios of sides. We have six trigonometric ratios Sin, Cos, Tan, Cosec, Sec, Cot.

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- sin A = Perpendicular / Hypotenuse
- cos A = Base / Hypotenuse
- tan A = Perpendicular / Base
- cot A = Base / Perpendicular
- sec A = Hypotenuse / Base
- cosec A = Hypotenuse / Perpendicular

Here, A is that the angle opposite to the perpendicular side.

Let’s see what the Perpendicular, base, and hypotenuse of a right triangle are,

**Perpendicular:**The side ahead of the angle is perpendicular. In this case, the side in front of 30 deg is called it’s perpendicular.**Base:**A base is one among the edges which contain an angle, except hypotenuse.**Hypotenuse:**It is a side opposite to 90°. it is the largest side.

Note:Perpendicular and base changes as angle changes. During a triangle, a side is perpendicular for an angle, but an equivalent side may be a base for an additional angle, but the hypotenuse remains an equivalent because it’s a side opposite to angle 90°.

In the above diagram for an equivalent triangle if considered angle 30° the perpendicular is that the side PQ, but if considered angle 60° the perpendicular is the side QR.

### If tan A = 4/3, then find the value of cos A

**Solution:**

Tan– The tan of an angle A is the ratio of lengths of perpendicular to the base.

Tan A = Perpendicular / Base

Cos –The cos of an angle A is the ratio of lengths of the base to the hypotenuse.

Cos A = Base / HypotenuseIn a right triangle if tan A is 4/3 it looks as follows.

tan A = 4/3

perpendicular / base = 4/3

Calculating hypotenuse –

H

^{2}= P^{2}+ B^{2}H

^{2}= 4^{2}+ 3^{2}H

^{2}= 16 + 9H

^{2}= 25H =5

Cos A = Base / hypotenuse

Cos A = 3/5

**Sample problem**

**Question 1: In a right angle triangle, angle A is 60°, and the base is 3m. Find the length of the** **hypotenuse.**

**Solution:**

Given: Base = 3m

Cos 60 = 1/2

B/H = 1/2

3/H = 1/2

H = 6

**Question 2: In a right angle triangle, angle A is 30°, and the Hypotenuse is 3m. Find the length of the Base.**

**Solution:**

Given: Hypotenuse = 3m

Cos 30° = √3/2

B/H = √3/2

B/3 = √3/2

B = 3√3/2

**Question 3: In a right angle triangle, for an angle A, the Perpendicular is 9√3m and the Base is 9m, find the angle A.**

**Solution:**

Given: perpendicular = 9√3m, Base = 9m.

Tan A = 9√3 / 9

Tan A = √3

Tan (60°) = √3

Angle A = 60°