In △OPQ, right-angled at P, OP=7 cm and OQ−PQ=1 cm. Determine the values of sinQ and cosQ.
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Step-by-Step Solution
Step 1: Set up the given information and equation
We are given a right-angled 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 △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=a2+2ab+b2. Here, a=1 and b=PQ. Also, calculate 72=49. Now, we simplify the equation.
Step 6: Isolate and solve for PQ
Subtract PQ2 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 sinQ
In △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 sinQ.
Step 9: Determine cosQ
For angle Q in △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 cosQ.