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Graphs of Trigonometric Functions Formula
The fundamental six functions of the Graphs Of Trigonometric Functions Formula have a range of numbers as their result and a domain input value that is the angle of a right triangle. The domain of the Graphs Of Trigonometric Functions Formula, often known as the “trigonometric function,” of f(x) = sin, is the angle, expressed in degrees or radians, and the range is [1, 1]. The domain and range of the other functions are similar. Calculus, geometry, and algebra all make heavy use of the Graphs Of Trigonometric Functions Formula. Students need to know the period, phase, amplitude, maximum and lowest turning points in order to sketch the Sine, Cosine, and Tangent trigonometry graphs. Numerous branches of Engineering and Science employ these Graphs Of Trigonometric Functions Formula.
What are Trigonometric Functions?
Trigonometry uses six fundamental trigonometric operations. Trigonometric ratios describe these operations. The sine function, cosine function, secant function, cosecant function, tangent function, and cotangent function are the six fundamental trigonometric functions. The ratio of sides of a rightangled triangle is the basis for trigonometric functions and identities. Using the Graphs Of Trigonometric Functions Formula, the sine, cosine, tangent, secant, and cotangent values are calculated for the perpendicular side, hypotenuse, and base of a right triangle.
Trigonometric Functions Formulas
The sides of a rightangled triangle can be used in the specific Graphs Of Trigonometric Functions Formula to get the values of trig functions. Students utilise the shortened version of these functions to write these Graphs Of Trigonometric Functions Formula. Sine is represented by the symbol sin, cosine by the symbol cos, tangent by the symbol tan, secant by the symbol sec, cosecant by the abbreviation cosec, and cotangent by the abbreviation cot. The following are the fundamental formulas for finding trigonometric functions:
 Perpendicular/hypotenuse = sin
 base/hypotenuse = cos
 hypotenuse/base cosec = hypotenuse/perpendicular
 cot = base/perpendicular
 tan = perpendicular/base sec = hypotenuse/base cosec
Sine and cosecant are reciprocals of one another, as students can see from the Graphs Of Trigonometric Functions Formula above. Cosine and secant, as well as tangent and cotangent, are the reciprocal pairs.
Trigonometric Functions Values
The domain of the Graphs Of Trigonometric Functions Formula is, which can be expressed in radians or degrees. These fundamental quantities, which are widely utilised in calculations, are also known as the standard values of trig functions at particular angles. Students can learn more about this topic with the assistance of the scholarly reference materials provided on the Extramarks educational website and the Extramarks Learning App. The Graphs Of Trigonometric Functions Formula fundamental values have been obtained from a unit circle. Additionally, all the Graphs Of Trigonometric Functions Formula are satisfied by these values. Sine, Cosine, and Tangent are the three primary trigonometric ratios on which trigonometry values are based.
 Sine or sin: Hypotenuse = BC or AC; Side opposite to
 sines or cosines A = Adjacent side to B and C = Hypotenuse
 Tangent or tan = Adjacent side to / side opposite to = BC / AB
The trigonometric values for reciprocal characteristics, Sec, Cosec, and Cot ratios can be expressed similarly.
Hypotenuse/adjacent side to angle = AC/AB; Sec = 1/Cos
Hypotenuse/side opposed to angle = AC/BC; Cosec = 1/Sin;
Cot = 1/tan = Side next to angle / Side across from angle = AB / BC
Also,
Sec θ. Cos θ =1
Cosec θ. Sin θ =1
Cot θ. Tan θ =1
Trig Functions in Four Quadrants
The angle is measured in the anticlockwise direction with reference to the positive xaxis and is an acute angle (90°). Additionally, depending on the positive or negative axis of the quadrant, these trig functions have various numeric signs (+ or ) in the various quadrants. In quadrants I and II, the Graphs Of Trigonometric Functions Formula of Sin and Cosec are positive; in quadrants III and IV, they are negative. The first quadrant of all Graphs Of Trigonometric Functions Formula has a positive range. Only Quadrants I and III have positive trigonometric functions Tan and Cot, and only Quadrants I and IV have positive trigonometric ratios Cos and Sec.
In the first quadrant, the trigonometric functions have values of, (90° – θ). The relationships between the several complimentary trigonometric functions for the angle (90° – θ) are provided by the cofunction identities.
 sin(90°−θ) = cos θ
 cos(90°−θ) = sin θ
 tan(90°−θ) = cot θ
 cot(90°−θ) = tan θ
 sec(90°−θ) = cosec θ
 cosec(90°−θ) = sec θ
The domain value for various Graphs Of Trigonometric Functions Formula is (π/2 + θ, π – θ), (π + θ, 3π/2 – θ) and (3π/2 + θ, 2π – θ) in the second, third, and fourth quadrants, respectively. The trigonometric values for π/2, and 3π/2 change according to their complementary ratios, such as Sinθ⇔Cosθ, Tanθ⇔Cotθ, Secθ⇔Cosecθ. The Graphs Of Trigonometric Functions Formula for π, 2π stay constant.
Trigonometric Functions Graph
The domain value is depicted on the horizontal xaxis of the Graphs Of Trigonometric Functions Formula, while the range value is represented along the vertical yaxis. While the graphs of other Graphs Of Trigonometric Functions Formula do not pass through the origin, the graphs of Sin and Tan do. Sin and Cos have a finite range of [1, 1].
Students need to know the period, phase, amplitude, maximum and lowest turning points in order to sketch the Sine, Cosine, and Tangent trigonometry graphs. Numerous branches of Engineering and Science employ these graphs.
Domain and Range of Trigonometric Functions
The resultant value is the Graphs Of Trigonometric Functions Formula range, while the value stands for the domain of trigonometric functions. The range is a real numerical value, while the domain values are in degrees or radians. In general, the trigonometric function’s domain is a real number value, however, in some circumstances, a few angle values are disregarded since they produce a range with an infinite value. Periodic functions make up the Graphs Of Trigonometric Functions Formula.
Trigonometric Functions Identities
Reciprocal identities, Pythagorean formulae, the sum and difference of trig function identities, formulas for multiple and submany angles, and sum and product of identities are the main categories of the Graphs Of Trigonometric Functions Formula. The ratio of sides of a rightangled triangle can be used to simply calculate all of the formulas below. The fundamental Graphs Of Trigonometric Functions Formula can be used to derive the higher formulas. Trigonometric issues are frequently made simpler by the application of reciprocal identities.
Reciprocal Identities
 cosec θ = 1/sin θ
 sec θ = 1/cos θ
 cot θ = 1/tan θ
 sin θ = 1/cosec θ
 cos θ = 1/sec θ
 tan θ = 1/cot θ
Pythagorean Identities
 Sin2θ + Cos2θ = 1
 1 + Tan2θ = Sec2θ
 1 + Cot2θ = Cosec2θ
Sum and 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)
HalfAngle Identities
 Sin A/2 is equal to [(1 – cos A) / 2].
 [(1 + cos A) / 2] cos A/2
 tan A/2 is calculated as [(1 – cos A) / (1 + cos A)] or sin A / (1 + cos A) (or) sin A / (1 – cos A)
Double Angle Identities
2sin = sin(2x) (x) cos(x) equals [2tan(x)/(1+tan2(x))]
[(1tan2 x)/(1+tan2 x)] cos(2x) = cos2(x)sin2(x)
2cos2 = cos(2x) (x)
−1 = 1–2sin2(x) (x)
[2tan(x)] = tan(2x)
/ [1−tan2(x)]
cot(2x) equals [cot2(x) – 1].
/[2cot(x)]
sec2 x = sec (2x) (2sec2 x)
cosec (2x) equals (sec x. cosec x)/2.
Triple Angle Identities
Sin 3x is equal to 3x – 4x.
Tan 3x equals [3 tan xtan3x]/[13tan2x]
Cos 3x = 4cos3x – 3cos x
Product identities
 2sinx⋅cosy=sin(x+y)+sin(x−y)
 2 cosx⋅cosy=cos(x+y)+cos(x−y)
 2sinxsiny=cos(x−y)−cos(x+y)
Sum of Identities
 sinx+siny=2 sin((x+y)/2) . cos(((x + y)/2)
 sinx + siny = 2cos((x + y)/2). sin((x−y)/2)
 Cos(x+y)/2) = cosx+cosy. cos((x−y)/2)
 Cosxcosy=sin((x+y)/2) and sin((xy)/2)
Inverse Trigonometric Functions
The inverse ratio of the Graphs Of Trigonometric Functions Formula is the basis for inverse trigonometric functions. Here, Sin1 x = can replace the fundamental Graphs Of Trigonometric Functions Formula of Sin = x. In this case, x can be expressed as a whole number, a decimal, a fraction, or an exponent. Students know that Equals Sin1 (1/2) for = 30°. Inverse Graphs Of Trigonometric Functions Formula can be created from any Graphs Of Trigonometric Functions Formula.
All six of the Graphs Of Trigonometric Functions Formula can be solved using the inverse Graphs Of Trigonometric Functions Formula for arbitrary values. The negative of the values is equivalent to the negatives of the function for the inverse Graphs Of Trigonometric Functions Formula of sine, tangent, and cosecant. Additionally, the negatives of the domain are equivalent to subtracting the function from the value for cosecant, secant, and cotangent functions.
 Sin1(x) = Sin1x
 Tan1(x) = Tan1(x)
 Cosec1(x) = Cosec1(x)
 Cos1(x) = Cos1(x)
 Sec1(x) = Sec1(x)
 Cot1(x) = Cot1(x)
Similar to the fundamental trigonometric functions are the inverse Graphs Of Trigonometric Functions Formula of reciprocal and complementary functions. The inverse Graphs Of Trigonometric Functions Formula can be understood in terms of the reciprocal relationship between the Graphs Of Trigonometric Functions Formula, sinecosecant, cossecant, and tangentcotangent. Additionally, sincecosine, tangentcotangent, and secantcosecant complimentary functions can be translated into:
Reciprocal Functions: The inverse sine, inverse cosine, and inverse tangent Graphs Of Trigonometric Functions Formula can alternatively be stated as follows.
Tan1x = Cot11/x Tan1x = Sec11/x Cosec11/x
supplementary roles: Sinecosine, tangentcotangent, and secantcosecant are complementary functions that add up to 2.
 Sin1x + Cos1x = π/2
 Tan1x + Cot1x = π/2
 Sec1x + Cosec1x = π/2
Trigonometric Functions Derivatives
The slope of the tangent to the curve is determined by the differentiation of the Graphs Of Trigonometric Functions Formula. By using the x value in degrees for Cosx and the differentiation of Sinx, the slope of the tangent to the Sinx curve at any given position. The equation of tangent, normal, and calculation errors can be found using the Graphs Of Trigonometric Functions Formula function differentiation.
d/dx. Sinx = Cosx
d/dx. Cosx = Sinx
d/dx. Tanx = Sec2x
d/dx. Cotx = Cosec2x
d/dx. Secx = Secx. Tanx
d/dx. Cosecx = – Cosecx. Cotx
Integration of Trigonometric Function
The integration of trigonometric functions is useful for determining the area under the Graphs Of Trigonometric Functions Formula. In general, the area under the trigonometric function’s graph can be calculated using any of the axis lines and up to a predetermined limit. The area of planar surfaces with irregular shapes may often be determined by integrating Graphs Of Trigonometric Functions Formula.
∫ cosx dx = sinx + C
∫ sinx dx = cosx + C
∫ sec2x dx = tanx + C
∫ cosec2x dx = cotx + C
∫ secx.tanx dx = secx + C
∫ cosecx.cotx dx = cosecx + C
∫ tanx dx = logsecx + C
∫ cotx.dx = logsinx + C
∫ secx dx = logsecx + tanx + C
∫ cosecx.dx = logcosecx – cotx + C
Solved Examples on Trigonometric Functions
 Find the Sin75° value.
Solution:
Finding Sin75° value is the goal.
The formula Sin(A + B) = SinA.CosB + CosA.SinB can be used here.
Here, A is 30 degrees and B is 45 degrees.
Sin(30 + 45) Equals Sin(75)
= Sin30°.
Cos45 plus Cos30.
Sin45°
= (1/2) (1/√2) + (√3/2) (1/√2)
= 1/2√2 + √3/2√2
= (√3 + 1) / 2√2
Sin75° = (3 + 1) / 22 is the solution.
 Determine the trigonometric functions’ values for the given value of 12Tan = 5.
Solution:
Given that 12Tan = 5, we can calculate Tan as 5/12.
Tan =Perpendicular/Base = 5/12.
With the Pythagorean theorem in use, we have:
Hypotenuse2 = Base2 + Perpendicular2
Hyp2 = 122 + 52
= 144 + 25
= 169
Hyp = 13
Consequently, the following are the other trigonometric functions.
Syntax: Sin = Perp/Hyp = 5/13
Base/Hyp Cos = 12/13
Base/Perp = 12/5 for Cot.
Hyp/Base = 13/12 for Sec.
Hyp/Perp + Cosec = 13/5
2. Determine the value of the six trigonometric functions’ product.
Students are aware that sec x is the reciprocal of cosec x and that sin x is its reciprocal. Additionally, cot x and tan x can be represented as the ratio of cos x and sin x, respectively. Thus,
Sinx, Cosx, Tanx, Cotx, Sec, and Cosec are all equal to Sinx, Cosx, (Sinx/Cosx), (Cosx/Sinx), and (1/Cosx) and Sinx, respectively.
= (sinx cosx) / (sinx cosx) / (cosx sinx)
= 1 × 1
= 1
The sum of the six trigonometric functions is 1, hence the answer is yes.
Practice Questions on Trigonometric Functions
 Show that a sin (B – C) + b sin (C – A) + c sin (A – B) = 0 for every triangle ABC.
Solution:
Any triangle ABC contains
a/sin A = b/sin B = c/sin C = k
a=k sin A, b=k sin B, and c=k sin C
LHS
= a sin (B – C) + b sin (C – A) + c sin (A – B)
= [sin B cos C – cos B sin C]k sin A [sin C cos A – cos C sin A] + k sin B [sin A cos B – cos A sin B] + k sin C
= (k sin A sin B cos C – k sin A cos B sin C) + (k sin B cos C sin A) + (k sin C cos A sin B)
= 0
= RHS
That a sin (B – C) + b sin (C – A) + c sin (A – B) = 0 is so proved.
2. In one minute, a wheel completes 360 revolutions. How many radians does it rotate around in a second?
Solution:
Given,
The wheel makes 360 revolutions in one minute.
60 seconds make up a minute.
There are six revolutions in a second, or 360/60.
360° is the angle made in one rotation.
Angles created in six revolutions are equal to six times 360 degrees.
The angle measured in radians over six revolutions is equal to 6 360 /180.
= 6 × 2 × π
= 12π
3. Demonstrate that tan 3x tan 2x tan x = tan 3x – tan 2x – tan x.
Solution:
Suppose 3x = 2x + x.
When applying “tan” to both sides,
Tan 3x equals Tan 2x + Tan x
Having said that,
(Tan 2x + Tan X)/tan 3x (1 tan 2x tan x)
Tan 3x = Tan 2x + Tan x = Tan 3x(1 – Tan 2x Tan X)
Tan 2x plus Tan X = Tan 2x – Tan 3x + Tan 2x
Tan 3x minus Tan 2x plus Tan X equals Tan 3x Tan 2x Tan X.
Tan 3x – Tan 2x – Tan X is equal to Tan 3x Tan 2x Tan X.
FAQs (Frequently Asked Questions)
1. What do trigonometric functions serve?
In geometric figures, unknown angles and distances are derived from known or measured angles using trigonometric functions. The need to calculate angles and distances in subjects like Astronomy, Cartography, Surveying, and Artillery Range Finding led to the development of Trigonometry.
2. What role do trigonometric functions have in practical life?
Trigonometry is used to establish directions like the north, south, east, and west. It also explains the direction to point the compass in order to travel straight forward. To locate a certain location, it is used in navigation. It is also employed to calculate the separation between a location in the sea and the shore.
3. How are trigonometric functions made?
The ratio of the sides of the rightangled triangle yields trigonometric functions. The six trigonometric ratios have the following values when the hypotenuse, base, altitude, and angle between the hypotenuse and base are the three sides of the triangle, respectively.