Given $\tan A = \dfrac{4}{3}$, find the other trigonometric ratios of the angle A.

Step 1: Define the given tangent ratio

We are given the value of $\tan A$. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. We can represent this with a right triangle $\triangle ABC$, where $\angle B$ is the right angle.

Step 2: Assign side lengths using a constant

From the ratio $\frac{BC}{AB} = \frac{4}{3}$, we can assume the lengths of the sides $BC$ and $AB$ are $4k$ and $3k$ respectively, where $k$ is a positive constant. This maintains the given ratio while allowing for scaling of the triangle.

Step 3: Calculate the hypotenuse using Pythagoras Theorem

Now we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This allows us to find the length of the hypotenuse, $AC$.

Step 4: Calculate $\sin A$

The sine of angle $A$ is the ratio of the length of the side opposite to angle $A$ to the length of the hypotenuse. We substitute the values we found for $BC$ and $AC$.

Step 5: Calculate $\cos A$

The cosine of angle $A$ is the ratio of the length of the side adjacent to angle $A$ to the length of the hypotenuse. We substitute the values we found for $AB$ and $AC$.

Step 6: Calculate $\text{cosec } A$

The cosecant of angle $A$ is the reciprocal of $\sin A$. We use the value of $\sin A$ calculated in the previous step.

Step 7: Calculate $\sec A$

The secant of angle $A$ is the reciprocal of $\cos A$. We use the value of $\cos A$ calculated in the previous step.

Step 8: Calculate $\cot A$

The cotangent of angle $A$ is the reciprocal of $\tan A$. We use the given value of $\tan A$.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

All Introduction to Trigonometry solutions →