In $\triangle OPQ$, right-angled at P, $OP = 7$ cm and $OQ - PQ = 1$ cm. Determine the values of $\sin Q$ and $\cos Q$.

Step 1: Set up the given information and equation

We are given a right-angled triangle $\triangle OPQ$ with the right angle at P. The length of side OP is 7 cm. We are also given a relationship between the lengths of the other two sides, OQ and PQ, which is $OQ - PQ = 1$ cm.

Step 2: Express OQ in terms of PQ

From the given equation $OQ - PQ = 1$, we can express the hypotenuse OQ in terms of the side PQ by adding PQ to both sides of the equation. This will be useful for substituting into the Pythagorean theorem.

Step 3: Apply the Pythagorean theorem

Since $\triangle OPQ$ is a right-angled triangle at P, we can apply the Pythagorean theorem. According to the theorem, the square of the hypotenuse (OQ) is equal to the sum of the squares of the other two sides (OP and PQ).

Step 4: Substitute the knowns and solve for PQ

Now we substitute the expression for OQ from step 2 and the given value of OP into the Pythagorean theorem. We then expand the equation and solve for the unknown side PQ. This will allow us to find the length of PQ.

Step 5: Expand and simplify the equation

Expand the term $(1 + PQ)^2$ using the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a=1$ and $b=PQ$. Also, calculate $7^2 = 49$. Now, we simplify the equation.

Step 6: Isolate and solve for PQ

Subtract $PQ^2$ from both sides of the equation. Then, subtract 1 from both sides to isolate the term with PQ. Finally, divide by 2 to find the value of PQ. So, the length of side PQ is 24 cm.

Step 7: Calculate OQ

Now that we have the value of PQ, we can find the length of OQ using the relationship $OQ = 1 + PQ$. Substitute $PQ = 24$ cm into this equation to get the value of OQ.

Step 8: Determine $\sin Q$

In $\triangle OPQ$, for angle Q, the side opposite to it is OP and the hypotenuse is OQ. We use the definition of sine as the ratio of the opposite side to the hypotenuse. Substituting the calculated values, we find $\sin Q$.

Step 9: Determine $\cos Q$

For angle Q in $\triangle OPQ$, the side adjacent to it is PQ and the hypotenuse is OQ. We use the definition of cosine as the ratio of the adjacent side to the hypotenuse. Substituting the calculated values, we find $\cos Q$.

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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