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Q- If tan B = 0.6, then find the value of sec^2 B + cosec^2 B.
Answers
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Answer:
We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts.
Pythagorean theorem: a2 + b2 = c2.
Sines: sin A = a/c, sin B = b/c.
Cosines: cos A = b/c, cos B = a/c.
Tangents: tan A = a/b, tan B = b/a.
Let’s first look at some cases where we don’t know all the sides. Suppose we don’t know the hypotenuse but we do know the other two sides. The Pythagorean theorem will give us the hypotenuse. For instance, if a = 10 and b = 24, then c2 = a2 + b2 = 102 + 242 = 100 + 576 = 676. The square root of 676 is 26, so c = 26. (It’s nice to give examples where the square roots come out whole numbers; in life
they usually don’t.)
Now suppose we know the hypotenuse and one side, but have to find the other. For example, if b = 119 and c = 169, then a2 = c2 – b2 = 1692 – 1192 = 28561 – 14161 = 14400, and the square root of 14400 is 120, so a = 120.
We might only know one side but we also know an angle. For example, if the side a = 15 and the angle A = 41°, we can use a sine and a tangent to find the hypotenuse and the other side. Since sin A = a/c, we know c = a/sin A = 15/sin 41. Using a calculator, this is 15/0.6561 = 22.864. Also, tan A = a/b, so b = a/tan A = 15/tan 41 = 15/0.8693 = 17.256. Whether you use a sine, cosine, or tangent depends on which side and angle you know.