Trigonometry Part 11 Extreme values, Maximum and Minimum of trigonometric functions. acosx+bsinx+c
Published at : October 06, 2021
Sin(90-x)=cosx
cos(90-x)=sinx
Tan(90-x)=cotx
Cotx(90-x)=tanx
sec(90°−x) = csc x
csc(90°−x) = sec x
Sum & Difference Identities
sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)
Double Angle Identities
sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
tan(2x) = [2tan(x)]/ [1−tan2(x)]
sec (2x) = sec2 x/(2-sec2 x)
csc (2x) = (sec x. csc x)/2
Triple Angle Identities
Sin 3x = 3sin x – 4sin3x
Cos 3x = 4cos3x-3cos x
Tan 3x = [3tanx-tan3x]/[1-3tan2x]
Half Angle Identities
sin
x
2
=
±
√
1
−
cos
x
2
cos
x
2
=
±
√
1
+
cos
x
2
tan
(
x
2
)
=
√
1
−
cos
(
x
)
1
+
cos
(
x
)
Also,
tan
(
x
2
)
=
√
1
−
cos
(
x
)
1
+
cos
(
x
)
=
√
(
1
−
cos
(
x
)
)
(
1
−
cos
(
x
)
)
(
1
+
cos
(
x
)
)
(
1
−
cos
(
x
)
)
=
√
(
1
−
cos
(
x
)
)
2
1
−
cos
2
(
x
)
=
√
(
1
−
cos
(
x
)
)
2
sin
2
(
x
)
=
1
−
cos
(
x
)
sin
(
x
)
So,
tan
(
x
2
)
=
1
−
cos
(
x
)
sin
(
x
)
Product identities
sin
x
⋅
cos
y
=
sin
(
x
+
y
)
+
sin
(
x
−
y
)
2
cos
x
⋅
cos
y
=
cos
(
x
+
y
)
+
cos
(
x
−
y
)
2
sin
x
⋅
sin
y
=
cos
(
x
−
y
)
−
cos
(
x
+
y
)
2
Sum to Product Identities
sin
x
+
sin
y
=
2
sin
x
+
y
2
cos
x
−
y
2
sin
x
−
sin
y
=
2
cos
x
+
y
2
sin
x
−
y
2
cos
x
+
cos
y
=
2
cos
x
+
y
2
cos
x
−
y
2
cos
x
−
cos
y
=
−
2
sin
x
+
y
2
sin
x
−
y
2
Inverse Trigonometry Formulas
sin-1 (–x) = – sin-1 x
cos-1 (–x) = π – cos-1 x
tan-1 (–x) = – tan-1 x
cosec-1 (–x) = – cosec-1 x
sec-1 (–x) = π – sec-1 x
cot-1 (–x) = π – cot-1 x
What is Sin 3x Formula?
Sin 3x is the sine of three times of an angle in a right-angled triangle, that is expressed as:
Sin 3x = 3sin x – 4sin3x
Trigonometry Formulas From Class 10 to Class 12
Trigonometry Formulas For Class 12
Trigonometry Formulas For Class 11
Trigonometry Formulas For Class 10
Trigonometry Formulas Major systems
All trigonometric formulas are divided into two major systems:
Trigonometric Identities
Trigonometric Ratios
Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle.
Here we provide a list of all Trigonometry formulas for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.
We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.
Solved Problems
Q.1:What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?
Sol: Given,
(sin30° + cos30°) – (sin 60° + cos60°)
= ½ + √3/2 – √3/2 – ½
= 0
Q.2: If cos A = 4/5, then tan A = ?
Sol: Given,
Cos A = ⅘
As we know, from trigonometry identities,
1+tan2A = sec2A
sec2A – 1 = tan2A
(1/cos2A) -1 = tan2A
Putting the value of cos A = ⅘.
(5/4)2 – 1 = tan2 A
tan2A = 9/16
tan A = 3/4
Frequently Asked Questions – FAQs
What are the basic trigonometric ratios?
Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.
What are formulas for trigonometry ratios?
Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base
What are the three main functions in trigonometry?
Sin, Cos and Tan are three main functions in trigonometry.
What are the fundamental trigonometry identities?
The three fundamental identities are:
1. sin2 A + cos2 A = 1
2. 1+tan2 A = sec2 A
3. 1+cot2 A = csc2 A
cos(90-x)=sinx
Tan(90-x)=cotx
Cotx(90-x)=tanx
sec(90°−x) = csc x
csc(90°−x) = sec x
Sum & Difference Identities
sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)
Double Angle Identities
sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
tan(2x) = [2tan(x)]/ [1−tan2(x)]
sec (2x) = sec2 x/(2-sec2 x)
csc (2x) = (sec x. csc x)/2
Triple Angle Identities
Sin 3x = 3sin x – 4sin3x
Cos 3x = 4cos3x-3cos x
Tan 3x = [3tanx-tan3x]/[1-3tan2x]
Half Angle Identities
sin
x
2
=
±
√
1
−
cos
x
2
cos
x
2
=
±
√
1
+
cos
x
2
tan
(
x
2
)
=
√
1
−
cos
(
x
)
1
+
cos
(
x
)
Also,
tan
(
x
2
)
=
√
1
−
cos
(
x
)
1
+
cos
(
x
)
=
√
(
1
−
cos
(
x
)
)
(
1
−
cos
(
x
)
)
(
1
+
cos
(
x
)
)
(
1
−
cos
(
x
)
)
=
√
(
1
−
cos
(
x
)
)
2
1
−
cos
2
(
x
)
=
√
(
1
−
cos
(
x
)
)
2
sin
2
(
x
)
=
1
−
cos
(
x
)
sin
(
x
)
So,
tan
(
x
2
)
=
1
−
cos
(
x
)
sin
(
x
)
Product identities
sin
x
⋅
cos
y
=
sin
(
x
+
y
)
+
sin
(
x
−
y
)
2
cos
x
⋅
cos
y
=
cos
(
x
+
y
)
+
cos
(
x
−
y
)
2
sin
x
⋅
sin
y
=
cos
(
x
−
y
)
−
cos
(
x
+
y
)
2
Sum to Product Identities
sin
x
+
sin
y
=
2
sin
x
+
y
2
cos
x
−
y
2
sin
x
−
sin
y
=
2
cos
x
+
y
2
sin
x
−
y
2
cos
x
+
cos
y
=
2
cos
x
+
y
2
cos
x
−
y
2
cos
x
−
cos
y
=
−
2
sin
x
+
y
2
sin
x
−
y
2
Inverse Trigonometry Formulas
sin-1 (–x) = – sin-1 x
cos-1 (–x) = π – cos-1 x
tan-1 (–x) = – tan-1 x
cosec-1 (–x) = – cosec-1 x
sec-1 (–x) = π – sec-1 x
cot-1 (–x) = π – cot-1 x
What is Sin 3x Formula?
Sin 3x is the sine of three times of an angle in a right-angled triangle, that is expressed as:
Sin 3x = 3sin x – 4sin3x
Trigonometry Formulas From Class 10 to Class 12
Trigonometry Formulas For Class 12
Trigonometry Formulas For Class 11
Trigonometry Formulas For Class 10
Trigonometry Formulas Major systems
All trigonometric formulas are divided into two major systems:
Trigonometric Identities
Trigonometric Ratios
Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle.
Here we provide a list of all Trigonometry formulas for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.
We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.
Solved Problems
Q.1:What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?
Sol: Given,
(sin30° + cos30°) – (sin 60° + cos60°)
= ½ + √3/2 – √3/2 – ½
= 0
Q.2: If cos A = 4/5, then tan A = ?
Sol: Given,
Cos A = ⅘
As we know, from trigonometry identities,
1+tan2A = sec2A
sec2A – 1 = tan2A
(1/cos2A) -1 = tan2A
Putting the value of cos A = ⅘.
(5/4)2 – 1 = tan2 A
tan2A = 9/16
tan A = 3/4
Frequently Asked Questions – FAQs
What are the basic trigonometric ratios?
Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.
What are formulas for trigonometry ratios?
Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base
What are the three main functions in trigonometry?
Sin, Cos and Tan are three main functions in trigonometry.
What are the fundamental trigonometry identities?
The three fundamental identities are:
1. sin2 A + cos2 A = 1
2. 1+tan2 A = sec2 A
3. 1+cot2 A = csc2 A
TrigonometryExtremevalues,