Trigonometric Functions
* Quadrilateral Angles: Angles that are integral multiples of \frac{π}{2} are called quadrilateral angles.
Some identities:
* Cos2θ + sin2 θ= 1
* Sec2 θ = 1 + tan²θ
* Cosec2θ = 1 + cot2θ
Sin (2nπ + θ) = sin θ, where n is any integer.
Cos (2nπ + θ) = cos θ, where n is any integer.
Sin θ = 0 θ ⇒ nπ, where n is any integer.
Cos θ = 0 θ ⇒ (2n + 1) \frac{π}{2}, where n is any integer
* In Quadrant I, all the trigonometric functions are positive.
* In Quadrant II, the sine and cosecant functions are positive.
* In Quadrant III, the tan and cot functions are positive.
* In Quadrant IV, the cos and secant functions are positive.
Angles
* An angle is a measure of rotation of a given ray about its initial point.
* An angle is of measure one degree if the rotation from the initial side to the terminal side is ( \frac{1}{360})th of a revolution.“
* A degree is divided into 60 equal parts, and each part is called a minute.
* A minute is further divided into 60 equal parts, and each part is called a second.
* The angle subtended by an arc at the centre of a circle whose length is equal to the radius of the circle, is 1 radian.
* In a circle of radius r, if an arc of length l subtends an angle θ radian at the centre,
then l = rθ.
Domain and Range of Trigonometric Functions
| Function | Domain | Range |
| sin θ | R | [ – 1 , 1 ] |
| cos θ | R | [ – 1 , 1 ] |
| tan θ | R – { (2n + 1) \frac{π}{2}, n ∈ Z } | R |
| cot θ | R – { nπ , n ∈ Z } | R |
| sec θ | R – { (2n + 1) \frac{π}{2} , n ∈ Z } | ( – ∞, – 1 ] U { 1 , ∞ ) or { y : y ∈ R, y ≥ 1 or y ≤ – 1 } |
| cosec θ | R – { nπ , n ∈ Z } | ( – ∞, – 1 ] U { 1 , ∞ ) or { y : y ∈ R, y ≥ 1 or y ≤ – 1 } |
Trigonometric Functions of Sum and Difference of Two Angles
(Part I)
Cos (x + y) = cos x cos y – sin x sin y
Cos (x – y) = cos x cos y + sin x sin y
Cos ( \frac{π}{2} – x) = sin x
Sin ( \frac{π}{2} – x) = cos x
Sin (x + y) = sin x cos y + cos x sin y
Sin (x – y) = sin x cos y – cos x sin y
Cos ( \frac{π}{2}+ x) = – sin x
Sin ( \frac{π}{2}+ x) = cos x
Tan (x + y) = \frac{tan \thinspace x + tan \thinspace y}{1 - tan \thinspace x \thinspace tan \thinspace y}, if x, y, x, + y ≠ (2n + 1) \frac{π}{2}
Tan (x – y) = \frac{tan \thinspace x - \thinspace tan \thinspace y}{1 + tan \thinspace x \thinspace tan \thinspace y}
Cot (x + y) = \frac{ cot \thinspace x \thinspace cot \thinspace y - 1}{cot \thinspace y \thinspace + \thinspace cot \thinspace x} , if x, y, x + y ≠ n
Cot (x – y) = \frac{ cot \thinspace x \thinspace cot \thinspace y + 1}{cot \thinspace y \thinspace - \thinspace cot \thinspace x}
Trigonometric Functions of Sum and Difference
(Part II)
Cos 2A = cos2A – sin2A = 2 cos2A – 1 =1 – 2 sin2A = \frac{ 1 \thinspace - \thinspace tan² \thinspace A}{ 1\thinspace + tan² \thinspace A}
Sin 2A = 2 sin A cos A = \frac{ 2 \thinspace tan \thinspace A}{ 1 \thinspace + tan² \thinspace A}
Tan 2A = \frac{ 2 \thinspace tan \thinspace A}{ 1 \thinspace - tan² \thinspace A}
Sin 3A = 3 sin A – 4 sin3A
Cos3A = 4 cos3A – 3 cos A
Tan 3A = \frac{ 3 tan \thinspace A - tan³ \thinspace A}{ 1 \thinspace - \thinspace 3tan² \thinspace A}
Sin (A +B) + sin (A – B) = 2 sin A cos B
Sin (A + B) – sin (A – B) = 2 cos A sin B
Cos (A +B) + cos (A – B) = 2 cos A cos B
Cos (A + B) – cos (A – B) = -2 sin A sin B
Sin C + sin D = 2 sin \frac{ C + D}{2}. cos \frac{ C - D}{2}
Sin C – sin D = 2 cos \frac{ C + D}{2}. sin \frac{ C - D}{2}
Cos C + cos D = 2 cos \frac{ C + D}{2}. cos \frac{ C - D}{2}
Cos C – cos D = -2 sin \frac{ C + D}{2}. sin \frac{ C - D}{2}
Sin (A + B) sin (A – B) = sin2A – sin2B = cos2B – cos2A
Cos (A + B) cos (A – B) = cos2A – sin2B = cos2B – sin2A
Tan ( \frac{π}{2} + A) = \frac{1 + tanA}{1 - tanA} = \frac{cosA + sinA}{cos A - sin A}
Trigonometric Equations
* An equation involving trigonometric functions of a variable is known as trigonometric equation.
* The value of the unknown angle that satisfies the given trigonometric equation is called the solution of the trigonometric equation.
* The solution of a trigonometric equation for which 0 ≤ x < 2 are called the principal solutions
* When the solution set of a trigonometric equation is an expression involving integer n, then such type of solution is called the general solution of the trigonometric equation.
* For any real number x and y, sin x = sin y implies x = n + (-1)² y, where n ∈ Z.
* For any real numbers x and y, cos x = cos y, implies x = 2n ± y, where n ∈ Z.
* If x and y are not odd multiplies of π/2, then tan x = tan y implies x = n + y, where n ∈ Z.