THE ULTIMATE TRIGONOMETRY LEARNING COMPILATION
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DEFINITION
DEFINITION
Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations.
It is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.
TOPICS
TOPICS
ANGLES IN STANDARD POSITIONS
ANGLES IN STANDARD POSITIONS
An angle is in standard position when its vertex is at the origin of the Cartesian coordinate plane and its initial side is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side. If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle.
REFERENCE ANGLES
REFERENCE ANGLES
Reference Angle Definition
The reference angle is the smallest possible angle made by the terminal side of the given angle with the x-axis. It is always an acute angle (except when it is exactly 90 degrees). A reference angle is always positive irrespective of which side of the axis it is falling.
How to Draw Reference Angle?
To draw the reference angle for an angle, identify its terminal side and see by what angle the terminal side is close to the x-axis. The reference angle of 135° is drawn below:
QUADRANTAL ANGLES
QUADRANTAL ANGLES
Quadrantal angles are angles in standard position whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°.
Examples
PYTHAGOREAN THEOREM
PYTHAGOREAN THEOREM
UNIT CIRCLE
UNIT CIRCLE
A unit circle from the name itself defines a circle of unit radius. A circle is a closed geometric figure without any sides or angles. The unit circle has all the properties of a circle, and its equation is also derived from the equation of a circle. Further, a unit circle is useful to derive the standard angle values of all the trigonometric ratios.
SOLVING RIGHT TRIANGLES
SOLVING RIGHT TRIANGLES
How to Solve a Right Triangle
Step 1: Determine which sides (adjacent, opposite, or hypotenuse) are known in relation to the given angle.
Step 2: Set up the proper equation with the trigonometric rule that relates the known side to the unknown side.
Step 3: Solve the equation for the unknown side.
DEGREES TO RADIAN
DEGREES TO RADIAN
Degrees and radians are the most commonly used measuring units to measure angles. A Degree is an angle made by one part of 360 equally divided parts of a circle at the centre with a radius of r. Radian is the angle made at the circle's centre by an arc of length equivalent to its radius.
Degrees to radians is a form of conversion used to convert the unit of measurement of angles in geometry. Just like every quantity has a unit of measurement, angles also are measured in different units depending upon the field of application. For measuring the angles, degrees and radians are used as the units of measurement. It is essential for one to know how to convert degrees to radians since most often, radians are used instead of degrees to measure angles.
AREA OF A SECTOR
AREA OF A SECTOR
The area of a sector of a circle is the amount of space occupied within the boundary of a sector of a circle. A sector always initiates from the center of the circle. The semi-circle is also a sector of a circle, in this case, a circle is having two sectors of equal size.
If the radius of the circle is (r) and the angle of the sector is (θ) is given, then the formula used to calculate the area of the sector is of
Area of sector (A) = (θ/360°) × πr2
θ is the angle in degrees.
r is the radius of the circle.
TRIGONOMETRIC FUNCTIONS
TRIGONOMETRIC FUNCTIONS
Trigonometric functions are also known as Circular Functions can be simply defined as the functions of an angle of a triangle. It means that the relationship between the angles and sides of a triangle are given by these trig functions. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.
TRIGONOMETRIC IDENTITIES
TRIGONOMETRIC IDENTITIES
Trigonometric identities are a fundamental aspect of trigonometry, which is the study of the relationships between the angles and sides of triangles. These identities are mathematical equations that involve trigonometric functions, such as sine, cosine, and tangent, and are true for all values of the variables involved.
PYTHAGOREAN IDENTITIES
PYTHAGOREAN IDENTITIES
The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. The following are the 3 Pythagorean trig identities.
sin2θ + cos2θ = 1. This can also be written as 1 - sin2θ = cos2 θ ⇒ 1 - cos2θ = sin2θ
sec2θ - tan2θ = 1. This can also be written as sec2θ = 1 + tan2θ ⇒ sec2θ - 1 = tan2θ
csc2θ - cot2θ = 1. This can also be written as csc2θ = 1 + cot2θ ⇒ csc2θ - 1 = cot2θ
COMPLEMENTARY ANGLE IDENTITIES
COMPLEMENTARY ANGLE IDENTITIES
The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90 - θ). The trigonometric ratios of complementary angles (also known as co-function Identities) are:
sin (90°- θ) = cos θ
cos (90°- θ) = sin θ
tan (90°- θ) = cot θ
cosec (90°- θ) = sec θ
sec (90°- θ) = cosec θ
cot (90°- θ) = tan θ
SUPPLEMENTARY ANGLE IDENTITIES
SUPPLEMENTARY ANGLE IDENTITIES
The supplementary angles are a pair of two angles such that their sum is equal to 180°. The supplement of an angle θ is (180 - θ). The trigonometric ratios of supplementary angles are:
sin (180°- θ) = sinθ
cos (180°- θ) = -cos θ
tan (180°- θ) = -tan θ
cosec (180°- θ) = cosec θ
sec (180°- θ)= -sec θ
cot (180°- θ) = -cot θ
SUM AND DIFFERENCE IDENTITIES
SUM AND DIFFERENCE IDENTITIES
The sum and difference identities include the formulas of sin(A+B), cos(A-B), tan(A+B), etc.
sin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B - cos A sin B
cos (A+B) = cos A cos B - sin A sin B
cos (A-B) = cos A cos B + sin A sin B
tan (A+B) = (tan A + tan B)/(1 - tan A tan B)
tan (A-B) = (tan A - tan B)/(1 + tan A tan B)
PERIODIC IDENTITIES
PERIODIC IDENTITIES
Periodic identities in trigonometry are a set of identities that describe the periodic nature of trigonometric functions. A periodic function is a function that repeats its values after a certain interval, known as its period. Here are the periodic identities of sin, cos, and tan.
sin(x + 2π) = sin(x)
cos(x + 2π) = cos(x)
tan(x + π) = tan(x)
DOUBLE ANGLE AND HALF ANGLE IDENTITIES
DOUBLE ANGLE AND HALF ANGLE IDENTITIES
Double angle formulas: The double angle trigonometric identities can be obtained by using the sum and difference formulas.
For example, from the above formulas:
sin (A+B) = sin A cos B + cos A sin B
Substitute A = B = θ on both sides here, we get:
sin (θ + θ) = sinθ cosθ + cosθ sinθ
sin 2θ = 2 sinθ cosθ
Half Angle Formulas: Using one of the above double-angle formulas,
cos 2θ = 1 - 2 sin2θ
2 sin2θ = 1- cos 2θ
sin2θ = (1 - cos2θ)/(2)
sin θ = ±√[(1 - cos 2θ)/2]
Replacing θ by θ/2 on both sides,
sin (θ/2) = ±√[(1 - cos θ)/2]
TRIPLE ANGLE IDENTITIES
TRIPLE ANGLE IDENTITIES
Triple angle identities are trigonometric identities that relate the values of trigonometric functions of three times an angle to the values of trigonometric functions of the angle itself.
SUM TO PRODUCT IDENTITIES
SUM TO PRODUCT IDENTITIES
Sum to Product Identities: These identities are
sin A + sin B = 2[sin((A + B)/2)cos((A − B)/2)]
sin A − sin B = 2[cos((A + B)/2)sin((A − B)/2)]
cos A + cos B = 2[cos((A + B)/2)cos((A − B)/2)]
cos A − cos B = −2[sin((A + B)/2)sin((A − B)/2)]
PRODUCT TO SUM IDENTITIES
PRODUCT TO SUM IDENTITIES
Product to Sum Identities: These identities are:
sin A⋅cos B = [sin(A + B) + sin(A − B)]/2
cos A⋅cos B = [cos(A + B) + cos(A − B)]/2
sin A⋅sin B = [cos(A − B) − cos(A + B)]/2
SINE LAW AND COSINE LAW
SINE LAW AND COSINE LAW
The trigonometric identities that we have learned are derived using right-angled triangles. There are a few other identities that we use in the case of triangles that are not right-angled.
Sine Rule: The sine rule gives the relation between the angles and the corresponding sides of a triangle. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule. For a triangle with sides 'a', 'b', and 'c' and the respective opposite angles are A, B, and C, the sine rule can be given as,
a/sinA = b/sinB = c/sinC
sinA/a = sinB/b = sinC/c
a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC
Cosine Rule: The cosine rule gives the relation between the angles and the sides of a triangle and is usually used when two sides and the included angle of a triangle are given. Cosine rule for a triangle with sides 'a', 'b', and 'c' and the respective opposite angles are A, B, and C, sine rule can be given as,
a2 = b2 + c2 - 2bc·cosA
b2 = c2 + a2 - 2ca·cosB
c2 = a2 + b2 - 2ab·cosC
INVERSE TRIGONOMETRIC FUNCTIONS
INVERSE TRIGONOMETRIC FUNCTIONS
Inverse trigonometric functions are defined as the inverse functions of the basic trigonometric functions, which are sine, cosine, tangent, cotangent, secant and cosecant functions. They are also termed arcus functions, antitrigonometric functions or cyclometric functions. These inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.
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