If (1+ sin t)(1 + cos t) = 5/4. Find the value of (1 - sin t)(1 – cos t).
Complete Trigonometry Chapter: https://www.youtube.com/playlist?list=PLQHaGos0gnrLjdYiY_F_IQ5XzAWFFqyLg If (1+ sin t)(1 + cos t) = 5/4. Find the value of (1 - sin t)(1 – cos t). Introduction Trigono + Metry = Triangle + Measure Triangle is any figure that has 3 angles. Angle is the measure of rotation of a line about a fixed point. Trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analyzing a musical tone and in many other areas. Parts of Angles: Initial side, Terminal Side & Vertex Two types of Angles: Positive & Negative Measure of angle is amount of rotation performed to get the terminal side from the initial side. Type of measure of angle: Degree Measure & Radian Measure Degree Measure 1° = 1/360th of rotation. ( 1 complete rotation = 360 degree) 1° = 60′ (1 degree =60 minutes) 1′ = 60″ ( 1 minute = 60 seconds) Lets draw 360°,180°, 270°, 410°, – 30°, 50° Radian Measure: 1 radian is the Angle subtended at centre by arc of length 1 unit in a circle of radius 1 unit. One revolution = 2π radian Relationship between Degree & Radian 2π radian =360o = One revolution Or 1 radian = 180°/ π = 57° 16′ approx. Also, 1° = (π / 180) radian = 0.01746 radian approx. Degree to Radian Formula Radian measure = (π/ 180) × Degree measure Degree measure = (180/π) x Radian measure Numerical: Convert 60° to radian form & Convert π into degree. Solution: Lets first convert 60° to radian Radian measure = (π/ 180) × Degree measure = (π/ 180) * 60° = π/3 radian Now let’s convert π into degree. Degree measure = (180/π) x Radian measure = 180/π * π = 180o Trigonometric Functions (Any Angle) In trigonometric ratios, we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions. You will observe that In Quadrant I, all Sinθ, Cosθ & Tan θ are all positive. In Quadrant II only Sinθ is positive In Quadrant III only Tanθ is positive In Quadrant IV only Cosθ is positive Signs of Cosec θ, Sec θ & Cot θ can easily be determined using signs of Sinθ, Cosθ & Tan θ respectively. Memory Tip to remember Signs: Add sugar to coffee If we rotate (clockwise or anticlockwise) line OP by 360o, it will come back to same position. Thus if θ increases (or decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. Thus, sin(2nπ + θ) = sinθ , n ∈ Z , cos(2nπ + θ) = cosθ, n ∈ Z tan(2nπ + θ) = tanθ, n ∈ Z Note that, in the above scenario, Sinθ = b/1 = b , cosθ =a/1 = a & tanθ = b/a. Also, in right Triangle POM , a2 + b2 =1 Using these 2 equations we can say that sin2 θ+ cos2 θ= 1 Also we can prove that 1 + tan2 θ = sec2 θ 1 + cot2 θ = cosec2 θ Domain & Range of Trigonometric Functions Trigonometric Functions sin (- x) = - sin x cos (- x) = cos x tan(-x) = - tan x cosec(-x) = -cosec x sec (-x) = sec x cot (-x) = -cot x cos (π/2-x) = sin x sin (π/2 –x )= cos x tan (π/2 –x )= cot x cot (π/2 –x )= tan x cos (π/2+ x ) = - sin x sin (π/2 +x) = cos x tan (π/2+ x ) = - cot x cot (π/2 +x) = - tan x cos (π – x) = - cos x sin (π – x) = sin x tan(π – x) = -tan x cos (π + x) = - cos x sin (π + x) = -sin x tan(π + x) = tan x 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 cos 2x = cos2x – sin2 x sin 2x = 2 sinx cos x tan 2x= 2tan x / (1- tan2 x) sin 3x = 3 sin x – 4 sin3 x cos 3x = 4 cos3 x – 3 cos x tan 3x = (3tan x – tan3 x)/(1-3 tan2 x) 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x 1- sin2 x = cos2 x 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 (x + y) – sin (x – y))/2 cos x + cos y = 2 cos (x+ y)/2 cos (x – y)/2 cos x – cos y = – 2sin (x+ y)/2 sin (x- y)/2 sin x + sin y = 2sin (x+ y)/2 cos(x- y)/2 sin x – sin y = 2cos (x+ y)/2 sin (x- y)/2 trigonometry class 11 for jee mains trigonometry class 11 trigonometry class 11 ncert trigonometry class 11 in hindi trigonometry class 11 all formulas trigonometry class 11 basics trigonometry class 11 cbse trigonometry class 11 concept trigonometry class 11 crash course trigonometry class 11 domain and range trigonometry class 11 derivations trigonometry class 11 full chapter trigonometry class 11 formulas tricks trigonometry class 11 for iit jee trigonometry class 11 graphs trigonometry class 11 general solution class 11 maths trigonometry graphs class 11 maths chapter trigonometry in hindi trigonometry class 11 iit jee
Complete Trigonometry Chapter: https://www.youtube.com/playlist?list=PLQHaGos0gnrLjdYiY_F_IQ5XzAWFFqyLg If (1+ sin t)(1 + cos t) = 5/4. Find the value of (1 - sin t)(1 – cos t). Introduction Trigono + Metry = Triangle + Measure Triangle is any figure that has 3 angles. Angle is the measure of rotation of a line about a fixed point. Trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analyzing a musical tone and in many other areas. Parts of Angles: Initial side, Terminal Side & Vertex Two types of Angles: Positive & Negative Measure of angle is amount of rotation performed to get the terminal side from the initial side. Type of measure of angle: Degree Measure & Radian Measure Degree Measure 1° = 1/360th of rotation. ( 1 complete rotation = 360 degree) 1° = 60′ (1 degree =60 minutes) 1′ = 60″ ( 1 minute = 60 seconds) Lets draw 360°,180°, 270°, 410°, – 30°, 50° Radian Measure: 1 radian is the Angle subtended at centre by arc of length 1 unit in a circle of radius 1 unit. One revolution = 2π radian Relationship between Degree & Radian 2π radian =360o = One revolution Or 1 radian = 180°/ π = 57° 16′ approx. Also, 1° = (π / 180) radian = 0.01746 radian approx. Degree to Radian Formula Radian measure = (π/ 180) × Degree measure Degree measure = (180/π) x Radian measure Numerical: Convert 60° to radian form & Convert π into degree. Solution: Lets first convert 60° to radian Radian measure = (π/ 180) × Degree measure = (π/ 180) * 60° = π/3 radian Now let’s convert π into degree. Degree measure = (180/π) x Radian measure = 180/π * π = 180o Trigonometric Functions (Any Angle) In trigonometric ratios, we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions. You will observe that In Quadrant I, all Sinθ, Cosθ & Tan θ are all positive. In Quadrant II only Sinθ is positive In Quadrant III only Tanθ is positive In Quadrant IV only Cosθ is positive Signs of Cosec θ, Sec θ & Cot θ can easily be determined using signs of Sinθ, Cosθ & Tan θ respectively. Memory Tip to remember Signs: Add sugar to coffee If we rotate (clockwise or anticlockwise) line OP by 360o, it will come back to same position. Thus if θ increases (or decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. Thus, sin(2nπ + θ) = sinθ , n ∈ Z , cos(2nπ + θ) = cosθ, n ∈ Z tan(2nπ + θ) = tanθ, n ∈ Z Note that, in the above scenario, Sinθ = b/1 = b , cosθ =a/1 = a & tanθ = b/a. Also, in right Triangle POM , a2 + b2 =1 Using these 2 equations we can say that sin2 θ+ cos2 θ= 1 Also we can prove that 1 + tan2 θ = sec2 θ 1 + cot2 θ = cosec2 θ Domain & Range of Trigonometric Functions Trigonometric Functions sin (- x) = - sin x cos (- x) = cos x tan(-x) = - tan x cosec(-x) = -cosec x sec (-x) = sec x cot (-x) = -cot x cos (π/2-x) = sin x sin (π/2 –x )= cos x tan (π/2 –x )= cot x cot (π/2 –x )= tan x cos (π/2+ x ) = - sin x sin (π/2 +x) = cos x tan (π/2+ x ) = - cot x cot (π/2 +x) = - tan x cos (π – x) = - cos x sin (π – x) = sin x tan(π – x) = -tan x cos (π + x) = - cos x sin (π + x) = -sin x tan(π + x) = tan x 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 cos 2x = cos2x – sin2 x sin 2x = 2 sinx cos x tan 2x= 2tan x / (1- tan2 x) sin 3x = 3 sin x – 4 sin3 x cos 3x = 4 cos3 x – 3 cos x tan 3x = (3tan x – tan3 x)/(1-3 tan2 x) 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x 1- sin2 x = cos2 x 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 (x + y) – sin (x – y))/2 cos x + cos y = 2 cos (x+ y)/2 cos (x – y)/2 cos x – cos y = – 2sin (x+ y)/2 sin (x- y)/2 sin x + sin y = 2sin (x+ y)/2 cos(x- y)/2 sin x – sin y = 2cos (x+ y)/2 sin (x- y)/2 trigonometry class 11 for jee mains trigonometry class 11 trigonometry class 11 ncert trigonometry class 11 in hindi trigonometry class 11 all formulas trigonometry class 11 basics trigonometry class 11 cbse trigonometry class 11 concept trigonometry class 11 crash course trigonometry class 11 domain and range trigonometry class 11 derivations trigonometry class 11 full chapter trigonometry class 11 formulas tricks trigonometry class 11 for iit jee trigonometry class 11 graphs trigonometry class 11 general solution class 11 maths trigonometry graphs class 11 maths chapter trigonometry in hindi trigonometry class 11 iit jee