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Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a right-angled triangle. It uses trigonometric ratios to find the missing or unknown angles or sides of a triangle.
Trigonometric ratios
โ Sine : The ratio of the opposite side to the hypotenuse โ โ Cosine : The ratio of the adjacent side to the hypotenuse โ โ Tangent : The ratio of the opposite side to the adjacent side โ โ Cosecant : The ratio of the hypotenuse to the opposite side โ โ Secant : The ratio of the hypotenuse to the adjacent side โ โ Cotangent : The ratio of the adjacent side to the opposite side
Trigonometric formulas
โ sin ฮธ = Opposite Side/Hypotenuse โ โ cos ฮธ = Adjacent Side/Hypotenuse โ โ tan ฮธ = Opposite Side/Adjacent Side โ โ sec ฮธ = Hypotenuse/Adjacent Side โ โ cosec ฮธ = Hypotenuse/Opposite Side โ โ cot ฮธ = Adjacent Side/Opposite
Trigonometric functions are also used to find the length of an arc of a circ
Hipparchus (180โ125 BC) is known as the "father of trigonometry".
He was a Greek astronomer, mathematician, and geographer.
Basic Trigonometry Formulas
S.no Property Mathematical value
1 sin A Perpendicular/Hypotenuse
2 cos A Base/Hypotenuse
4 sec A 1/cos A
MathsTrigonometry Sign Functions
โ sin (-ฮธ) = โ sin ฮธ โ โ cos (โฮธ) = cos ฮธ โ โ tan (โฮธ) = โ tan ฮธ โ โ cosec (โฮธ) = โ cosec ฮธ โ โ sec (โฮธ) = sec ฮธ โ โ cot (โฮธ) = โ cot ฮธ
Trigonometric Identities
โ sin2A + cos2A = 1 โ โ tan2A + 1 = sec2A โ โ cot2A + 1 = cosec2A
โ sin (3ฯ/2 + ฮธ) = โ cos ฮธ โ โ cos (3ฯ/2 + ฮธ) = sin ฮธ โ โ sin (2ฯ โ ฮธ) = โ sin ฮธ โ โ cos (2ฯ โ ฮธ) = cos ฮธ
Sum and Difference of Two Angles
โ sin (A + B) = sin A cos B + cos A sin B โ โ sin (A โ B) = sin A cos B โ cos A sin B โ โ cos (A + B) = cos A cos B โ sin A sin B โ โ cos (A โ B) = cos A cos B + sin A sin B โ โ tan(A + B) = [(tan A + tan B) / (1 โ tan A tan B)] โ โ tan(A โ B) = [(tan A โ tan B) / (1 + tan A tan B)]
Double Angle Formulas
โ sin 2A = 2 sin A cos A = [2 tan A /(1 + tan2A)] โ โ cos 2A = cos2A โ sin2A = 1 โ 2 sin2A = 2 cos2A โ 1 = [(1 โ tan2A)/(1 + tan2A)] โ โ tan 2A = (2 tan A)/(1 โ tan2A)
Triple Angle Formulas
โ sin 3A = 3 sinA โ 4 sin3A โ โ cos 3A = 4 cos3A โ 3 cos A โ โ tan 3A = [3 tan A โ tan3A] / [1 โ 3 tan2A]
Solved Examples Using Important Trigonometry
Formulas
Q1. If cot Q = tan P then prove that P + Q = 90ยฐ.
Ans. Given,
tan P = cot Q
As we know, cot(90ยฐ โ A) = Tan A.
So, cot Q = cot(90ยฐ โ P)
Therefore, Q = 90ยฐ โ P
And
P + Q = 90ยฐ
Hence proved.
Application to science
While these developments shifted trigonometry away from its original connection to triangles, the practical aspects of the subject were not neglected. The 17th and 18th centuries saw the invention of numerous mechanical devicesโfrom accurate clocks and navigational tools to musical instruments of superior quality and greater tonal
