Half angle formula trigonometry class 11

Introduction Trigonometric identities are relationships between quantities (angles) that are likely for all values of the involved variables. Geometrically speaking, these identities are of one or more angles or perhaps even half or quarter of an angle. There are several trigonometric identities. First and foremost is the Pythagorean theorem, which involves a circle of unit one. x2+y2=1, whose trigonometric counterpart happens to be sin2+cos2=1. From the Pythagorean theorem for the unit circle, it can be thus said that: sin=√1-cos2 cos=√1-sin2, where the symbol + or – depends on the quadrant of the circle the angle theta is in. There are addition and subtraction identities, followed by multiple angle formulae and power of angle formulae. Simple calculations Suppose you are asked to calculate the value of 7.5 degrees using half-angle formulae for sine, cosine, and tangent. You know that 7.5 is the half of 15 degrees and that 15 degrees fall under the half of 30 degrees. Now, you already know the value of 30 degrees. The half angle formulae equations are as follows: sin2(/2)=(1-cos)/2 cos2(/2)=(1+cos)/2 tan(/2)=(1-cos )sin =sin (1+cos ) The tangent’s half-angle formula does not require a plus or a minus sign, unlike those for the sine or the cosine. The function for sine can be first calculated as 30/2 and then 15/2. The same goes for cosine and tangent. The Area of a Right-Angled Triangle using Trigonometry As we already know, 12baseheight gives the area of a right triangle...

Hint: In this problem, we have to find the value of \[\tan 22.5\] using a half -angle formula. We will use the half angle formula for tangent as follows. \[\tan 2\theta = \dfrac\theta \]

\text$, the only unknown value is the adjacent side length. This can be calculated by using the pythagrean theorem which states: \text=0.043619387365336$.

Pythagorean Identities Signs of sin, cos, tan in different quadrants To learn sign of sin, cos, tan in different quadrants, we remember A dd → S ugar → T o → C offee Representing as a table Quadrant I Quadrant II Quadrant III Quadrant IV sin + + – – cos + – – – tan + – + – Radians Radian measure = π/180 × Degree measure Also, 1 Degree = 60 minutes i.e. 1° = 60’ 1 Minute = 60 seconds i.e. 1’ = 60’’ Negative angles (Even-Odd Identities) sin (–x) = – sin x cos (–x) = cos x tan (–x) = – tan x sec (–x) = sec x cosec (–x) = – cosec x cot (–x) = – cot x Value of sin, cos, tan repeats after 2π sin (2π + x) = sin x cos (2π + x) = cos x tan (2π + x) = tan x Shifting angle by π/2, π, 3π/2 (Co-Function Identities or Periodicity Identities) sin (π/2 – x) = cos x cos (π/2 – x) = sin x sin (π/2 + x) = cos x cos (π/2 + x) = – sin x sin (3π/2 – x) = – cos x cos (3π/2 – x) = – sin x sin (3π/2 + x) = – cos x cos (3π/2 + x) = sin x sin (π – x) = sin x cos (π – x) = – cos x sin (π + x) = – sin x cos (π + x) = – cos x sin (2π – x) = – sin x cos (2π – x) = cos x sin (2π + x) = sin x cos (2π + x) = cos x Angle sum and difference identities Double Angle Formulas Triple Angle Formulas Half Angle Identities (Power reducing formulas) Sum Identities (Sum to Product Identities) Product Identities (Product to Sum Identities) Product to sum identities are 2 cos⁡x cos⁡y = cos⁡ (x + y) + cos⁡(x - y) -2 sin⁡x sin⁡y = cos⁡ (x + y) - cos⁡(x - y) 2 sin⁡x cos⁡y = sin⁡ (x + y) + sin⁡(x - y) 2 cos⁡x sin⁡y = sin⁡ ...

Trigonometry For Class 11 Trigonometry is one of the major topics in Maths subject. Trigonometry deals with the measurement of angles and sides of a triangle. Usually, trigonometry is considered for the right-angled triangle. Also, its functions are used to find out the length of the arc of a circle, which forms a section in the circle with a radius and its center point. If we break the word trigonometry, ‘Tri’ is a Greek word which means ‘Three’, ‘Gon’ means ‘length’, and ‘metry’ means ‘measurement’. So basically, trigonometry is a study of triangles, which has angles and lengths on its side. Trigonometry basics consist of sine, cosine and tangent functions. Trigonometry for class 11 contains trigonometric functions, identities to solve complex problems more simply. Trigonometry Formulas Here, you will learn trigonometry formulas for class 11 and trigonometric functions of Sum and Difference of two angles and trigonometric equations. Starting with the basics of Trigonometry formulas , for a right-angled triangle ABC perpendicular at B, having an angle θ, opposite to perpendicular (AB), we can define trigonometric ratios as; Sin θ = P/H Cos θ = B/H Tan θ = P/B Cot θ = B/P Sec θ = H/B Cosec θ = H/P Where, P = Perpendicular B = Base H = Hypotenuse Trigonometry Functions Trigonometry functions are measured in terms of radian for a circle drawn in the XY plane. Radian is nothing but the measure of an angle, just like a degree. The difference betw...

Hint: In this problem, we have to find the value of \[\tan 22.5\] using a half -angle formula. We will use the half angle formula for tangent as follows. \[\tan 2\theta = \dfrac\theta \]

Introduction Trigonometric identities are relationships between quantities (angles) that are likely for all values of the involved variables. Geometrically speaking, these identities are of one or more angles or perhaps even half or quarter of an angle. There are several trigonometric identities. First and foremost is the Pythagorean theorem, which involves a circle of unit one. x2+y2=1, whose trigonometric counterpart happens to be sin2+cos2=1. From the Pythagorean theorem for the unit circle, it can be thus said that: sin=√1-cos2 cos=√1-sin2, where the symbol + or – depends on the quadrant of the circle the angle theta is in. There are addition and subtraction identities, followed by multiple angle formulae and power of angle formulae. Simple calculations Suppose you are asked to calculate the value of 7.5 degrees using half-angle formulae for sine, cosine, and tangent. You know that 7.5 is the half of 15 degrees and that 15 degrees fall under the half of 30 degrees. Now, you already know the value of 30 degrees. The half angle formulae equations are as follows: sin2(/2)=(1-cos)/2 cos2(/2)=(1+cos)/2 tan(/2)=(1-cos )sin =sin (1+cos ) The tangent’s half-angle formula does not require a plus or a minus sign, unlike those for the sine or the cosine. The function for sine can be first calculated as 30/2 and then 15/2. The same goes for cosine and tangent. The Area of a Right-Angled Triangle using Trigonometry As we already know, 12baseheight gives the area of a right triangle...

Pythagorean Identities Signs of sin, cos, tan in different quadrants To learn sign of sin, cos, tan in different quadrants, we remember A dd → S ugar → T o → C offee Representing as a table Quadrant I Quadrant II Quadrant III Quadrant IV sin + + – – cos + – – – tan + – + – Radians Radian measure = π/180 × Degree measure Also, 1 Degree = 60 minutes i.e. 1° = 60’ 1 Minute = 60 seconds i.e. 1’ = 60’’ Negative angles (Even-Odd Identities) sin (–x) = – sin x cos (–x) = cos x tan (–x) = – tan x sec (–x) = sec x cosec (–x) = – cosec x cot (–x) = – cot x Value of sin, cos, tan repeats after 2π sin (2π + x) = sin x cos (2π + x) = cos x tan (2π + x) = tan x Shifting angle by π/2, π, 3π/2 (Co-Function Identities or Periodicity Identities) sin (π/2 – x) = cos x cos (π/2 – x) = sin x sin (π/2 + x) = cos x cos (π/2 + x) = – sin x sin (3π/2 – x) = – cos x cos (3π/2 – x) = – sin x sin (3π/2 + x) = – cos x cos (3π/2 + x) = sin x sin (π – x) = sin x cos (π – x) = – cos x sin (π + x) = – sin x cos (π + x) = – cos x sin (2π – x) = – sin x cos (2π – x) = cos x sin (2π + x) = sin x cos (2π + x) = cos x Angle sum and difference identities Double Angle Formulas Triple Angle Formulas Half Angle Identities (Power reducing formulas) Sum Identities (Sum to Product Identities) Product Identities (Product to Sum Identities) Product to sum identities are 2 cos⁡x cos⁡y = cos⁡ (x + y) + cos⁡(x - y) -2 sin⁡x sin⁡y = cos⁡ (x + y) - cos⁡(x - y) 2 sin⁡x cos⁡y = sin⁡ (x + y) + sin⁡(x - y) 2 cos⁡x sin⁡y = sin⁡ ...

Trigonometry For Class 11 Trigonometry is one of the major topics in Maths subject. Trigonometry deals with the measurement of angles and sides of a triangle. Usually, trigonometry is considered for the right-angled triangle. Also, its functions are used to find out the length of the arc of a circle, which forms a section in the circle with a radius and its center point. If we break the word trigonometry, ‘Tri’ is a Greek word which means ‘Three’, ‘Gon’ means ‘length’, and ‘metry’ means ‘measurement’. So basically, trigonometry is a study of triangles, which has angles and lengths on its side. Trigonometry basics consist of sine, cosine and tangent functions. Trigonometry for class 11 contains trigonometric functions, identities to solve complex problems more simply. Trigonometry Formulas Here, you will learn trigonometry formulas for class 11 and trigonometric functions of Sum and Difference of two angles and trigonometric equations. Starting with the basics of Trigonometry formulas , for a right-angled triangle ABC perpendicular at B, having an angle θ, opposite to perpendicular (AB), we can define trigonometric ratios as; Sin θ = P/H Cos θ = B/H Tan θ = P/B Cot θ = B/P Sec θ = H/B Cosec θ = H/P Where, P = Perpendicular B = Base H = Hypotenuse Trigonometry Functions Trigonometry functions are measured in terms of radian for a circle drawn in the XY plane. Radian is nothing but the measure of an angle, just like a degree. The difference betw...

\text$, the only unknown value is the adjacent side length. This can be calculated by using the pythagrean theorem which states: \text=0.043619387365336$.