Understanding Trigonometric Ratios: When Tan A 1/2

In the world of trigonometry, understanding the relationships between different trigonometric ratios plays a crucial role. This article explores how to determine the values of cos A and sin A given that the tangent of angle A is equal to 1/2.

Using the Definition of Tangent

Let's start by using the definition of tangent. The tangent of an angle is the ratio of the sine to the cosine:

tan A sin A / cos A

Given that tan A 1/2, we can write:

sin A (1/2) cos A

Utilizing the Pythagorean Identity

To find the exact values of sin A and cos A, we can use the Pythagorean identity, which states:

sin^2 A cos^2 A 1

Substituting sin A (1/2) cos A into the identity:

((1/2) cos A)^2 cos^2 A 1

(1/4) cos^2 A cos^2 A 1

(1/4) cos^2 A (4/4) cos^2 A 1

(5/4) cos^2 A 1

cos^2 A 4/5

cos A ±(2/√5) ±(2√5/5)

Determining the Sign of Sin A and Cos A

The signs of sin A and cos A depend on the quadrant in which angle A lies. However, for the purpose of this calculation, we have:

sin A ±(√5/5)

Verification Using a Right-Angled Triangle

For a clearer understanding, consider a right-angled triangle where the opposite side to angle A is of length 1 and the adjacent side is of length 2. Using the Pythagorean theorem:

Hypotenuse^2 1^2 2^2 5

Hypotenuse √5

The sine of angle A is the ratio of the opposite side to the hypotenuse:

sin A 1/√5 √5/5

The cosine of angle A is the ratio of the adjacent side to the hypotenuse:

cos A 2/√5 2√5/5

Checking the Results Using the Pythagorean Identity

We can verify our results by checking that the sum of the squares of the sine and cosine of angle A equals 1:

(√5/5)^2 (2√5/5)^2

(5/25) (4×5/25) 5/25 20/25 25/25 1

To summarize, when tan A 1/2, the values of sin A and cos A are:

Remember that the signs of these values depend on the specific quadrant of the angle A.