Question: Given a right triangle PQR, where m∠Q = 90°, PQ = 12 cm and PR = 13 cm, find the value of tan P - cot R.
fig. 1
Explanation
To find the value of tan P−cot R, we need to calculate the values of tan P and cot R using △PQR.
From the formulas that we learnt in the notes of basics of trigonometry, we get
tan P=Adjacent side of ∠POpposite side of ∠P=PQQR−−−(1)
Similarly, doing the same for cot R, we get
cot R=Opposite side of ∠RAdjacent side of ∠R=PQQR−−−(2)
From equation (1) and equation (2), we will require the values of side QR and side PQ. But as we do not have the value of QR, we'll first apply Pythagoras Theorem on △PQR and find the value of QR.
Solution
We know that △PQR is a right angled triangle, hence applying Pythagoras Theorem we get
PQ2+QR2=PR2
Putting the values of PQ and PR in the above equation to find the value of
∴ ∴ ∴ ∴ 122+QR2=132QR2=132−122QR2=169−144QR2=25QR2=52∴ QR=5−−−(3)
Now that we have the value of QR, we can find the value of equation that is asked in the question
===tan P−cot RPQQR−PQQR125−125 0tan P−cot R=0
Bonus section
Quick tip #1 (to find the value of the problem)
Sometimes these kinds of questions can be asked for only 1 mark as MCQ, so it might not be required to show such a long calculation. So here's a short explanation on how to quickly solve such kinds of questions.
You can start from what is asked in the question, like so:
=tan P−cot RPQQR−PQQR
Let's assume the value of PQQR to be x. Hence we get the equation as:
==PQQR−PQQR x−x 0
Quick tip #2 (to calculate the third side of a right angled triangle)
There is a term in Mathematics known as Pythagorean triples. A pythagorean triple is a set of three positive integers that can be written in the form of a2+b2=c2. Some of the well known Pythagorean triples are
- (3, 4, 5)
- (6, 8, 10)
- (5, 12, 13)
For any Pythagorean triple, the addition of squares of the two smallest values would be equal to the square of the largest value in that triple. And that is what the equation a2+b2=c2 says.
Whenever you are dealing with a right angled triangle where two of its sides are known and are in a need to find the third side, try to solve it using Pythagorean triples.
In the question above, we have the largest side, hypotenuse, as PR (13 cm) and one other side PQ (12 cm). We can directly apply the Pythagorean triple (5, 12, 13) and know that the remaining side of the triangle i.e. QR will be 5 cm.
Note that, you can apply Pythagorean triple only when dealing with a right angled triangle.
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