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Trignometry Functions of Sum and difference of angles in trigonometry help in calculating the height of the mountain, and the distance between Earth and various planets. Trigonometric identities are used to calculate long distances. These are the list of equations that will be used to establish the relationship between the sum and difference of angles in trigonometry functions. These functions are cos(-x) = cos and Sin (-x) = sin.
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Key Terms: Trigonometric Functions, Trigonometric identities, Height, Distance, Triangles, Sum and Difference of Angles
Trigonometry Functions
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Trigonometric functions are used to determine the relationship between the angles of triangles. It is also known as circular functions. Trigonometry functions include sin, cos, cosec, tan, sec, and cot. They are also known as trigonometric ratios.
There are various trigonometry formulas that help in representing the relation between the trigonometric identities and help them to find the unknown angle of a triangle.
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Relation Between Sum and Difference of Angles in Trigonometry Functions
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The functional values of any angle can be calculated by the sum and differences of angles in trigonometry functions. However, it is widely used to find the exact values of the angles and to establish a relationship between them using the sum and difference of various sin, cos, and tan of angles 30°, 45°,60°,90°,180°, 270°, and 360°.
Trigonometric Functions Detailed Video Explanation
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Trigonometry Functions of Sum and Difference of Angles Formulas
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The sum and Difference of angles formulas are tabulated below:
| Sum of Two Trigonometry Angles Identity | Difference between Two Trigonometry Angles Identity |
|---|---|
| Sin (Ѳ+A) = sin Ѳ cos A + cos Ѳ sin A | Sin (Ѳ -A) = sin Ѳ cos Q- cos Ѳ sin Q |
| Cos (Ѳ +A) = cos Ѳ cos A - sin Ѳ sin A | Cos (Ѳ -A) = cos Ѳ cos A + sin Ѳsin A |
| Tan ( Ѳ + A) = tan Ѳ + tan A/ 1- tan Ѳ tan A | Tan ( Ѳ+ A) = tan Ѳ - tan A/ 1- tan Ѳ tan A |
Read More: Area of a Triangle
Trigonometric Identities
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Trigonometric Identities are equalities that use trigonometry functions and hold true for all the values of the specified variables in the equation. There are numerous distinctive trigonometric identities that relate to a triangle's side length and angle.
- Only the right-angle triangle is consistent with the trigonometric identities.
- The six trigonometric ratios serve as the foundation for all trigonometric identities.
- These are sine, cosine, tangent, cosecant, secant, and cotangent.
- The adjacent side, opposite side, and hypotenuse side of the right triangle are used to define each of these trigonometric ratios.
- The six trigonometric ratios are the source of all fundamental trigonometric identities.
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Trigonometric Functions of Sum and Differences of Angles
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In the below figure:
Trigonometric Functions of Sum and Differences of Angles
A circle is drawn with the center as the origin and radius 1 unit. A point P1 is chosen at an angle of x units from the x-axis. The coordinates are mentioned in the figure. Another point P2 is chosen, at an angle of y units from the line segment OP1. P3 is a point on the circle which is at an angle of y units from the x-axis, measured clockwise.
Now, in the given figure, Δ OP1P3 is congruent to Δ OP2P4, by SAS congruency criteria.
Hence, P1P3 = P2P4 (CPCT)
⇒(P1P3 )2= (P2P4)2
Since we know the coordinates of all four points, hence using the distance formula, we can write:
[cos x – cos (-y)]2 + [sin x – sin (-y)]2 = [1- cos (x+y)]2 + sin2 (x+y)
On solving the above equation, we have the following identity:
Now if we will substitute suitable values in the above equation (1), (2), (5), and (6), we will have the following equation:
- Cos (\(\pi/2\)+ x) = - sin x
- Sin (\(\pi/2 \)+ x)= cos x
- Cos (\(\pi\) ±) x) = -cos x
- Sin (\(\pi\)- x) = sin x
- Sin (\(\pi\)+ x) = - sin x
- Sin (\(\pi/2 \) - x) = - sin x
- Cos (\(2\pi\) - x)= cos x
After grasping the concept of the expanded form of the trigonometric functions of sum and difference of angles of sin and cos, the expansion of tan and cot can be derived by,
- Tan(\(\alpha\) + A) = (tan α + tan A)/ (1-tan α tan A)
- Tan(\(\alpha\) - A) = (tan α - tan A)/ (1+ tan α tan A)
Similarly, we can calculate the following
- Cot (α + A) = (cot α cot A -1 )/ (cot A + cot α)
- Cot α - A) = (cot α cot A+ 1 )/ (cot A - cot α)
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Things to Remember
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- Trigonometric Functions of sum and differences of angles illustrate the relation between angles of triangles.
- The functional values of any angle can be calculated using circular functions.
- To calculate the relation between trigonometric identities, the sum, and the difference is used.
- A trigonometric identity is an equality that holds true for all possible values of the stated variables in the equation.
- Trigonometric function is also referred to as the circular function.
Previous Years’ Questions
- The value of sin2 51∘+ sin2 39∘ is… [KCET – 2020]
- If cos x = |sin x| then, the general solution is… [KCET – 2019]
- If 0≤ x< π/2, then the number of values of x… [JEE Main – 2019]
- A vertical lamp-post at the midpoint D… [JEE Main – 2019]
- If tanA+cotA=2, then the value of… [KCET – 2020]
- The value of cos245∘−sin215∘ is… [KCET – 2017]
- A, B and C are the angles opposite to the corresponding sides of lengths… [JKCET – 2017]
- The value of tan 8/π is equal to… [KCET – 2016]
- The value of tan10∘ tan20∘ tan30∘ tan40∘ tan50∘ tan60∘… [COMEDK UGET – 2012]
- A value of θ satisfying sin5θ−sin3θ+sinθ… [KCET – 2011]
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Sample Questions
Ques. Prove cos (30 + Ѳ) = √3/2 cosΘ -sinΘ/2 (5 Marks)
Ans. Using the formula,
Cos (Ѳ + A) = cos Ѳ cos A - sin Ѳ sin A , and using 30° - 60° angle,
First, we will solve the left-hand side (LHS),
LHS = Cos(30° + Ѳ)
= Cos 30° cos Ѳ - sin 30° sin Ѳ
= 3-√3/2 cos Ѳ - ½ sin Ѳ
Since LHS = RHS
Hence proved.
Ques. Show that cos(π /2 + Ѳ) = -sin Ѳ (3 Marks)
Ans. By using the formula
Cos ( Ѳ + A) = cos Ѳ cos A - sin Ѳ sin A
cos(π /2 + Ѳ) = cos π/2 cos Ѳ- sin π/2 sin Ѳ
= 0 * cos Ѳ -1 * sin Ѳ
= - sin Ѳ
Since LHS = RHS
Hence, proved
Ques. Find the value 5 sin 30â° + 3 tan 45° (2 Marks)
Ans. Given, 5sin30°+ 3tan45° = 5 × ½ + 3 × 1
=5/2 + 3
=5 + 6/2
=11/2
To solve the above equation we have substituted the value of angles of sin and tan.
Ques. Find the value of 2 sin2 30° tan 60° – 3 cos2 60° sec2 30° (3 Marks)
Ans. 2(1/2)2 × √3-3(1/2)2 × (2/√3)2
=2 × ¼ × √3-3 × ¼ × 4/3
=√3/2-1
=(√3-2)/2
The above equation can be solved by equating the values of the angles of sin, cos, tan, and sec and then simplifying the equation.
Ques. In a right triangle ABC right angle at B the six trigonometric ratios of ∠C. (5 Marks)
Ans. sinA=Perpendicular/Hypotenuse=3/5
Base=√((Hypotenuse)2-(Perpendicular)2)
=√(52-32 )
=√(25-9)=√16=4
Now
sinC = 4/5 = BC/AC, cosecC = 5/4
cosC = 3/5 = AB/AC, secC = 5/3
tanC = 4/3 = AB/AC, cotC = 3/4
Ques. Find the value of 2 sin2 30° tan 60° – 3 cos2 60° sec2 30° (2 Marks)
Ans. 2(1/2)2 × √3-3(1/2)2 × (2/√3)2
= 2 × ¼ × √3-3 × ¼ × 4/3 =√3/2-1 = (√3-2)/2
The above equation can be solved by equating the values of the angles of sin, cos, tan, and sec and then simplifying the equation.
Ques. Find the value of x. Tan 3x = sin 45° cos 45° + sin 30° (3 Marks)
Solution: tan3 x = 1/\(\sqrt2\) × 1/\(\sqrt2\) + 1/2
= ½ + ½ = 1
⇒tan3x = 1 ⇒ tan3x = tan45°
3x = 45°
X = 15°
The above equation can be solved by equating the values of the sin, tan, and cos and then using the formula of tan3x.
Ques. A 25 m long ladder is placed against a vertical wall of a building. The foot of the ladder is 7m from the base of the building. If the top of the ladder slips 4m, then the foot of the Ladder will slide by how much distance. (3 Marks)
Ans. Let the height of the wall be h.
Now, h = √(252-72 )
= √(576 ) = 24m
QS = \(\sqrt(625-400)\)
= \(\sqrt(225 )\)=15m
Required distance, X = (15-7) = 8m
Ques. What are the three main functions of trigonometry? (2 Marks)
Ans. The three main functions of trigonometry are Sine, Cosine, and Tangent.
Sin θ = Opposite / Hypotenuse
Cos θ = Adjacent/Hypotenuse
Tan θ = Opposite/Adjacent
Ques. Find the value θ sin2θ=√3 (2 Marks)
Ans. sin2θ= √3/2
2θ = 60
θ = 30°
The above equation can be solved by equating the θ.
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