Consider $\triangle ACB$, right-angled at C, in which $AB = 29$ units, $BC = 21$ units and $\angle ABC = \theta$. Determine the values of (i) $\cos^2 \theta + \sin^2 \theta$, (ii) $\cos^2 \theta - \sin^2 \theta$.

Step 1: Identify given values and the unknown side

We are given a right-angled triangle $\triangle ACB$, where the right angle is at C. We know the lengths of the hypotenuse $AB = 29$ units and the side $BC = 21$ units. We need to find the length of the side $AC$ to calculate the trigonometric ratios. We can use the Pythagorean theorem for this.

Step 2: Calculate the length of side AC

Substitute the given values into the Pythagorean theorem to calculate the length of side $AC$. We have $AB = 29$ and $BC = 21$. Therefore, $AC = \sqrt{29^2 - 21^2} = \sqrt{841 - 441} = \sqrt{400} = 20$ units. This value will be used to determine the sine and cosine of $\theta$.

Step 3: Determine $\sin \theta$ and $\cos \theta$

For angle $\theta$ (which is $\angle ABC$), the side opposite is $AC$, and the side adjacent is $BC$. The hypotenuse is $AB$. Using the definitions of sine and cosine for a right-angled triangle, we find $\sin \theta = \frac{AC}{AB} = \frac{20}{29}$ and $\cos \theta = \frac{BC}{AB} = \frac{21}{29}$. These values are crucial for solving both parts of the problem.

Step 4: Calculate $\cos^2 \theta + \sin^2 \theta$

Substitute the calculated values of $\sin \theta$ and $\cos \theta$ into the expression $\cos^2 \theta + \sin^2 \theta$. This fundamental trigonometric identity states that for any angle $\theta$, $\cos^2 \theta + \sin^2 \theta = 1$. Our calculation confirms this identity for the given triangle.

Step 5: Calculate $\cos^2 \theta - \sin^2 \theta$

Substitute the calculated values of $\sin \theta$ and $\cos \theta$ into the expression $\cos^2 \theta - \sin^2 \theta$. Perform the squaring and subtraction operations to find the final numerical value. This gives us $\frac{41}{841}$.

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).

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