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In mathematics, trigonometry refers to the sides and angles of a right-angle triangle. It is derived from the Greek words trigonon, which means triangle, and metron, which means measure.
- The hypotenuse, base, and perpendicular are three sides of a right-angle triangle.
- Hypteneuse is the largest side of the triangle.
- Trigonometric ratios are defined as the ratio between the sides of a triangle based on the angle formed between them.
- There are three types of ratios in trigonometry, namely, sine, cosine, and tangent ratios.
- All the other ratios are derived from these primary ratios.
- The trigonometric ratios are used in the fields of mechanics, geometry, and other science-related subjects.
Key Ratios: Trigonometry, Trigonometric Ratios, Right Angled Triangle, Triangle, Sine, Cosine, Tangent, Trigonometric Ratio Table, Angles
Trigonometric Ratios
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Trigonometric ratios are used to calculate the ratios of any two sides of a right-angled triangle on the basis of respective angles. The three sides of a triangle are opposite, adjacent, and hypotenuse.
- Perpendicular is the opposite side of the right-angled triangle.
- The base is the side of the triangle that is adjacent to the angle.
- It is measured with respect to the positive x-axis in the anticlockwise direction.
Practical applications of trigonometric ratios include electronics, electrical engineering, astronomy, geography, oceanography, seismology, and phonetics.
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How to find Trigonometric Ratios?
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The sin, cos, tan, cot, cosec, and secs are six different type of trigonometric ratios. Consider a right angle triangle with angle C and perpendicular side as B.
Sine
Sine is defined as the ratio of perpendicular side to hypotenuse angle. Hence,
sin C = (Opposite side to ∠C)/(Hypotenuse) = AB/AC
Cosine
Cosine is defined as the ratio of the adjacent side to the hypotenuse angle is the Cos angle.
cos C = (Adjacent side to ∠C)/(Hypotenuse) = BC/AC
Tangent
Tangent is defined as the ratio of opposite side angle to the adjacent side. Hence,
tan C = (Opposite side to ∠C)/(Adjacent side to ∠C) = AB/BC
Cosec
Cosec is an inverse of sin angle.So,
cosec C= 1/sin C = (Hypotenuse)/ ( Opposite side to ∠C) = AC/AB
Sec
The sec angle is inverse of cosine. Hence,
sec C = 1/cos C = (Hypotenuse)/ (Opposite side to ∠C) = AC/BC
Cot
The inverse of tangent is cotangent. So,
cot C = 1/tan C = (Adjacent side to ∠C)/(Opposite side to ∠C)= BC/AB
- These are the various ratios for triangle ABC.
- Angle B is 90° and angle C and A are 60° and 30°.
- The acute angles ratios are the same for every right-angled triangle.
Trigonometric Ratios Table
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Several mathematical calculations are necessary for trigonometric ratios. The table given below shows the ratios of various angles like 0 °, 30 °, 45 °, 60 °, and 90°.
| Angle | 0 ° | 30 ° | 45 ° | 60 ° | 90° |
|---|---|---|---|---|---|
| Sin C | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| Cos C | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| Tan C | 0 | 1/√3 | 1 | √3 | ∞ |
| Cosec C | ∞ | 2 | √2 | 2/√3 | 1 |
| Sec C | 1 | 2/√3 | √2 | 2 | ∞ |
| Cot C | ∞ | √3 | 1 | 1/√3 | 0 |
Example of Trigonometric Ratios TableExample: The trigonometric ratio for the above value angles are,
|
Trigonometry Applications
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Trigonometry is crucial branch of mathematics. Some applications of trigonometry are helpful in daily life which are as follows:
- Trigonometry helps to calculate the height of mountains and towers.
- The applications are used in the aviation industry and satellite systems.
- It determine the distance between sea and shore.
- The ratio is used to calculate the mechanical and electromagnetic waves quantities.
- The output power of solar panels is determined by the trigonometric ratio.
- It helps with creation of map.
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Things to Remember
- Trigonometric Ratios refers to all the values of all the trigonometric functions based on which the ratio of sides ofright-angled triangle is calculated.
- A right triangle hypotenuse, base, and perpendicular sides are used to determine a ratio.
- Sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (secs) are several trigonometric ratios
- Mnemonics determines relationships in different trigonometric functions.
- Pythagorean theorem, the sum-to-product and product-to-sum formulas, and DeMoivre's theorem are methods used to prove trigonometric identities.
Sample Questions
Ques: Consider a right-angled triangle ABC. Here, AC = 5cm, BC = 3cm and AB = 4cm. Find tan θ, sin θ, and cos θ If the angle of triangle ABC is θ? (3 marks)
Ans: In the triangle ABC,
Hypotenuse, AC = 5cm
Base, BC = 3cm
Perpendicular, AB = 4cm
So, tan θ = Perpendicular/Base = 4/3
Sin θ = Perpendicular/Hypotenuse = AB/AC = 4/5
Cos θ = Base/Hypotenuse = BC/AC = 3/5
Hence, these are the angles for triangle ABC in respect to tan, sin, and cos.
Ques: Consider sin θ = 12/5 and cos θ = 3/5. Find the value of tan θ? (3 marks)
Ans: The values given here are,
sin θ = 12/5
cos θ = 3/5.
So, Tan θ = Sin θ/Cos θ
= (12/5)/(3/5)
= 12/3
= 4
Hence, the value of tan θ is 4.
Ques: How does the cos value increases from 0° to 90°? (2 marks)
Ans. According to the trigonometric table ratio, cos 0° = 1, cos 90° = 0. So, the value of θ rises from 0 to 90°. Hence, the value of cos θ lowers from 1 to 0.
Ques: Consider a triangle ABC with right angle B. Find the value of Sin (A + C)? (3 marks)
Ans. Angle B = 90°
Angle A+B+C = 180°
So, Angle A+C+90 = 180
Angle A+C = 90°
Therefore, sin (A + C) = sin 90° = 1
Ques: Consider sin (A + B) = 1 and tan (A – B) = 1/√3. Find the values of tan A + cot B and sec A – cosec B? (4 marks)
Ans. The value given here is sin (A + B) = 1. We know that Sin 90°=1. So,
Sin (A + B) = Sin 90°
(A + B) = 90° - Equation 1
Here, tan (A – B) = 1/√3. Hence,
tan (A – B) = tan 30°
(A – B) = 30° - Equation 2
From the equation 1 and 2, we depict that, A = 60° and B= 30°. So,
tan A + cot B = tan 60° + cot 30°
= √3+ √3 = 2√3
Also, sec A – cosec B = sec 60° - cosec 30°
= 2-2 = 0
Ques: A building is at a distance of 180 feet from point A on the ground. Find the height of the building if tan θ = 4/9? (2 marks)
Ans: The triangle formed is a right-angled triangle. Now apply the trigonometric ratio of tanθ to calculate the height of the building.
tan θ = Perpendicular/Base
4/9 = Height/180 ft
Height = (4 × 180/9) = 80 ft
Ques: Consider sin θ = 18/5 and cos θ = 6/5. Find the value of tan θ? (3 marks)
Ans: The values given here are,
sin θ = 18/5
cos θ = 6/5.
So, Tan θ = Sin θ/Cos θ
= (18/5)/(6/5)
= 18/6
= 3
Hence, the value of tan θ is 3
Ques: In a right-angled triangle ABC, which is right-angled at B, we have AB = 8, and BC = 6. Then find sin A and tan A, cos C and cot C? (5 marks)
Ans: AC2=((AB)2 +(BC)2 )
=((6)2 +82 )
=(36 + 64 )
=100
=10
When we consider the t-ratios of∠A we have
Base AB = 8
Perpendicular BC = 6
Hypotenuse AC = 10
sinA= Perpendicular/Hypotenuse= 6/10
tanA= Perpendicular/Base= 6/8
When we consider t-ratios of ∠C, we have
Base BC = 6
Perpendicular AB = 8
Hypotenuse AC = 10
cosC = Base/Hypotenuse = 6/10
cotC = Base/Perpendicular = 6/8
Ques: Reema sees a bird sitting on the branch of a tree at an angle of elevation of 30°. Find the height at which the bird is sitting if Riya is standing 60 miles away from the tree? (3 marks)
Ans. Let us assume a right triangle ABC in which A is the position of the bird, B is the tree touching the ground, and C is the position of Anjali.
Thus,
- BC = 60 miles
- Angle C = 30°
- AB = x miles
Tan C = Opposite Side/Adjacent Side
tan(30°) = x/60
x = 60 × tan(30°)
x = 60 × 0.57 = 34.2
Ques: A building is at a distance of 450 feet from point A on the ground. Find the height of the building if tan θ = 3/5? (2 marks)
Ans: The triangle formed is a right-angled triangle. Now apply the trigonometric ratio of tanθ to calculate the height of the building.
tan θ = Perpendicular/Base
3/5 = Height/450 ft
Height = (3 × 450/5) = 270 ft
Ques: Consider sin θ = 22/5 and cos θ = 11/5. Find the value of tan θ? (3 marks)
Ans: The values given here are,
sin θ = 22/5
cos θ = 11/5.
So, Tan θ = Sin θ/Cos θ
= (22/5)/(11/5)
= 2
Hence, the value of tan θ is 2
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