Exams Prep Master
Trigonometry is the study of triangle side ratios. Trigonometry has six fundamental ratios: sin, cos, tan, cot, sec, and cosec. The formulas for each of these ratios differ. It makes use of a right-angled triangle's three sides and angles. Trigonometry is one of the most important branches of mathematics, with numerous applications in a wide range of fields. Using trigonometric formulas, functions, or trigonometric identities, it is possible to find the missing or unknown angles or sides of a right triangle. Angles in trigonometry can be measured in degrees or radians. 0°, 30°, 45°, 60°, and 90° are some of the most commonly used trigonometric angles for calculations. Let’s discuss this in detail along with the cosec cot formula.
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Also Read: First Order Differential Equation
Trigonometric Ratios
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Do you measure the height of a tall building if you're standing in front of it? Without going to the top of the building, you can use trigonometry concepts to determine its height. Trigonometry concepts can be used to calculate the width of a river. This is a triangle with a right angle.
Angle C is represented by, which is a common symbol in trigonometry. Our task here is to locate. The hypotenuse is the side opposite the 90 degrees, and the base is the side containing the 90 degrees and angle. The perpendicular is the side that is perpendicular to the angle. Now we can calculate the side ratio, which is known as a trigonometric ratio. Sin is the perpendicular to hypotenuse ratio. The hypotenuse refers to cos as the base. Tan is perpendicular to the base. Cosec is defined as the inverse of sin.
As a result, it will be 1/sin, and the ratio will be hypotenuse by perpendicular. Sec is defined as the reciprocal of cos, 1/cos, and the ratio is hypotenuse by the base. Cot is defined as 1/tan, and the base to perpendicular ratio is base by perpendicular.
Sin θ = P/H
Cos θ = B/H
Tan θ = P/B
Cosec θ = 1/sin θ = H/P
Sec θ = 1/cos θ = H/B
Cot θ = 1/tan θ = B/P
Also read: Isosceles Triangle Theorems
Trigonometry Table
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| 0 degree | 30 degree | 45 degree | 60 degree | 90 degree | |
|---|---|---|---|---|---|
| Sin θ | 0 | 1/2 | 1/ √2 | √3/2 | 1 |
| Cos θ | 1 | √3/2 | 1/ √2 | 1/2 | 0 |
| Tan θ | 0 | 1/ √3 | 1 | √3 | ∞ |
Read more: Independent Events in Probability
Trigonometric Identities
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Trigonometric Identities come in handy when trigonometric functions are used in an expression or equation. Trigonometric identities hold true for any value of a variable that appears on both sides of an equation. Geometrically, these identities involve one or more trigonometric functions (such as sine, cosine, and tangent). The primary trigonometry functions are sine, cosine, and tangent, while the other three functions are cotangent, secant, and cosecant. All six trig functions are used to derive the trigonometric identities. Trigonometric Identities are equalities that involve trigonometry functions and hold true for all variables in the equation.
There are numerous trigonometric identities involving the side length and angle of a triangle. Only the right-angle triangle has the trigonometric identities. Some basic trigonometric identities include:
Sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = cosec2 θ
Tan θ = sin θ/cos θ
Cot θ = cos θ/sin θ
Cosec Cot Formula
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Let's take a look at the Cosec Cot Formula.
Cosec x is given for an acute angle x in a right triangle by Cosec x = Hypotenuse / Opposite side.
The value of cot x is given by,
Adjacent Side / Opposite Side = Cot x
The following is the Cosec Cot Formula:
1+cot2θ=cosec2θ
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Applications of Trigonometry
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You've probably wondered how we know the heights of tall mountains, ancient monuments, and skyscrapers. All of this is made possible by trigonometric concepts.
Assume you want to know the height of a nearby building and you're 20 feet away from it. Assume your height is five feet tall. Look up at the top of the building and make a mental note of the angle your line of sight makes with the horizontal. All of your trigonometric concepts can be applied wherever there are right angle triangles. Assume the angle you've noted is 30 degrees. Tan 30 can be defined as the opposite side of the adjacent side.
If you're wondering how to find the angle of inclination, there are several widely used instruments available, including the clinometer/inclinometer, theodolite, and goniometer. Trigonometry is also used extensively in criminology. As a result, trigonometry is extremely useful in blood to spatter analysis. That is simply the interpretation of blood stain patterns. This is primarily done by forensic experts in order to reconstruct the scene of a crime or accident and to determine where the blood came from.
So, if the width of a blood stain is 8 mm and the length is 16 mm, they can calculate the angle of impact. That angle of impact now tells us the angle at which the blood struck the surface. In this case, you will also get a right angle triangle and can apply trigonometric concepts.
Read more: Multiplication Theorem on Probability
Things to Remember
- The study of triangle side ratios is known as trigonometry. There are six fundamental ratios in trigonometry: sin, cos, tan, cot, sec, and cosec.
- Some of the most commonly used trigonometric angles for calculations are 0°, 30°, 45°, 60°, and 90°.
- Trigonometric Identities are equalities that involve trigonometry functions and hold true for all variables in the equation.
- The Cosec Cot Formula is given as follows: 1+cot2θ=cosec2θ.
- Anywhere there are right angle triangles, you can apply all of your trigonometric concepts.
Solved Questions
Ques: Demonstrate that (cosec – cot)2 = (1 – cos )/(1 + cos ). (4 marks)
Ans: LHS = (cosec – cot)2 is the solution.
= (1 / sin cos/sin ) 2
= (((1cos )/sin ) 2
RHS = (1 – cos)/(1 + cos).
By rationalising the numerator,
= (1cos )/(1+cos )(1cos )(1cos )(1cos )
= (1cos)2/(1cos2 )
(1cos )2/sin2 =
= (((1cos )/sin )2
As a result, LHS = RHS.
Ques: If Tan P = 4 / 3, find Cot P. (3 marks)
Ans: Using the Cotangent formula, we can deduce that
Tan P = 1 / Cot P = 1.
= 1/ (4/3)
3/4 =
As a result, Cot P = 3/4.
Ques: Find the value of Cosec x if Sin x = 3/5. (1 mark)
Ans: 1/sinx = 5/3 cosec x
Ques: Find the value of Cosec x if Sin x = 5/7. (1 mark)
Ans: 7/5 cosec x = 1/sinx
Answer: 7/5
Ques: Demonstrate that (cosec θ – cot θ)2 = (1 – cos θ)/(1 + cos θ). (3 marks)
Ans: LHS = (cosec θ – cot θ)2
=(1sin θ−cos θsin θ)2=(1−cos θsin θ)2=(1sin θ−cos θsin θ)2=(1−cos θsin θ)2RHS = (1 – cos θ)/(1 + cos θ)
By rationalising the denominator,
=1−cos θ1+cos θ×1−cos θ1−cos θ=(1−cos θ)21−cos2θ=(1−cos θ)2sin2θ=(1−cos θsin θ)2=1−cos θ1+cos θ×1−cos θ1−cos θ=(1−cos θ)21−cos2θ=(1−cos θ)2sin2θ=(1−cos θsin θ)2
Therefore, LHS = RHS
Hence, proved that (cosec θ – cot θ)2 = (1 – cos θ)/(1 + cos θ).
Ques: Simplify the expression below. (2 marks)
Ans: √(1−sin2A)cosec2A(1−sin2A)cosec2A
Solution:
√(1−sin2A)cosec2A=√cos2A×cosecA=√cos2Asin2A=√cot2A=cotA
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