Trigonometry

Important Trigonometry Identities

  1. sin\,\theta=\frac{P}{H}cos\,\theta=\frac{B}{H};   tan\,\theta=\frac{P}{B}
  2.  cosec\,\theta=\frac{1}{sin\,\theta}
  3. tan\,\theta=\frac{1}{cot\,\theta}
  4. cos\,\theta=\frac{1}{sec\,\theta}
  5. tan\,\theta=\frac{sin\,\theta}{cos\,\theta}
  6. cot\,\theta=\frac{cos\,\theta}{sin\,\theta}
  7.  sin^{2}\,\theta+ cos^{2}\,\theta=1
  8.  1+cot^{2}\,\theta=cosec^{2}\,\theta
  9.  1+tan^{2}\,\theta=sec^{2}\,\theta

 

 0^{o}  30^{o}  45^{o}  60^{o}  90^{o}
sin\theta 0 \frac{1}{2} \frac{1}{\sqrt{2}} \frac{\sqrt{3}}{2} 1
cos\theta 1 \frac{\sqrt{3}}{2} \frac{1}{\sqrt{2}} \frac{1}{2} 0
tan\theta 0 \frac{1}{\sqrt{3}} 1 {\sqrt{3}} Infinite

 


  1.  sin(-\theta)= -sin\theta
  2.  cos(-\theta)= cos\,\theta
  3.  tan(-\theta)= -tan\,\theta
  4.  cot(-\theta)= -cot\,\theta
  5.  sec(-\theta)= sec\,\theta
  6.  cosec(-\theta)= -cosec\,\theta

trigonometry

a. (90-\theta)\to  FIRST QUADRANT

b. (90+\theta)\to  SECOND QUADRANT

c. (180+\theta)\to  THIRD QUADRANT

d. (180-\theta)\to  SECOND QUADRANT

e. (270+\theta)\to FOURTH QUADRANT

f. (270-\theta)\to THIRD QUADRANT


  1.  sin(90-\theta)= cos\,\theta
  2.  cos(90-\theta)= sin\,\theta
  3.  tan(90-\theta)= cot\,\theta
  4.  cot(90-\theta)= tan\,\theta
  5.  sec(90-\theta)= cosec\,\theta
  6.  cosec(90-\theta)= sec\,\theta

  1. sin(90+\theta)= +cos\,\theta
  2. cos(90+\theta)= -sin\,\theta
  3. tan(90+\theta)= -cot\,\theta
  4. cot(90+\theta)= -tan\,\theta
  5. sec(90+\theta)= -cosec\,\theta
  6. cosec(90+\theta)= +sec\,\theta

  1.  sin(180-\theta)= +sin\,\theta
  2. cos(180-\theta)= -cos\,\theta
  3. tan(180-\theta)= -tan\,\theta
  4. cot(180-\theta)= -cot\,\theta
  5. sec(180-\theta)= -sec\,\theta
  6. cosec(180-\theta)= +cosec\,\theta

  1.  sin(180+\theta)= -sin\,\theta
  2. cos(180+\theta)= -cos\,\theta
  3. tan(180+\theta)= +tan\,\theta
  4. cot(180+\theta)= +cot\,\theta
  5. sec(180+\theta)= -sec\,\theta
  6. cosec(180+\theta)= -cosec\,\theta

  1. sin(A+B) = sinAcosB + cosAsinB
  2. sin(A-B) = sinAcosB – cosAsinB
  3. cos(A+B) = cosAcosB – sinAsinB
  4. cos(A-B) = cosAcosB + sinAsinB

  1. tan (A + B) = \frac{tanA+tanB}{1-tanAtanB}
  2. tan (A – B) = \frac{tanA-tanB}{1+tanAtanB}
  3. cot (A + B) = \frac{cotA\,cotB-1}{cotA+cotB}
  4. cot (A – B) = \frac{cotA\,cotB+1}{cotB-cotA}

  1. sin (A+B) sin (A-B) = sin ^{2}\,A-sin ^{2}\,B = cos ^{2}\,B-cos ^{2}\,A
  2. cos (A+B) cos (A-B) = cos ^{2}\,A-sin^{2}\,B = cos ^{2}\,B-sin^{2}\,A

  1.  sin2\,\theta=2sin\theta\,cos\theta=\frac{2tan\theta}{1+tan^{2}\,\theta
  2.  cos2\,\theta=2cos^{2}\theta-1=cos^{2}\theta-sin^{2}\theta =1-2sin^{2}\theta=\frac{1-tan^{2}\theta}{1+tan^{2}\theta}
  3.  tan2\,\theta=\frac{2tan\theta}{1-tan^{2}\,\theta

  1. sin3\.\theta=3sin\.\theta-4 sin^{3}\theta
  2. cos3\.\theta=4 cos^{3}\theta-3cos\.\theta
  3. tan3\theta=\frac{3tan\theta- tan^{3}\theta}{1-3 tan^{2}\theta}

  1. -1\leq sin\theta\leq 1
  2. -1\leq cos\theta\leq 1
  3. -\infty\leq tan\theta\leq \infty

Maximum and Minimum value of a trigonometrical function of the form asin\theta+bcos\theta are \sqrt{a ^{2}+b ^{2}} and -\sqrt{b ^{2}+a ^{2}} respectively.

For Example:1.  asin\theta-bcos\theta

Maximum Value= \sqrt{a ^{2}+b ^{2}}

2. Find the maximum and minimum value of 8sin\,\theta\,cos\,\theta+4 cos2\theta

Solution: This can be written as:

\Rightarrow 4(2sin\,\theta\,cos\,\theta)+4 cos2\theta

\Rightarrow 4sin2\theta+4 cos2\theta

Maximum Value \Rightarrow \sqrt{ 4^{2}+ 4^{2}}=4\sqrt{2}

Minimum Value \Rightarrow-\sqrt{ 4^{2}+ 4^{2}}=-4\sqrt{2}


  1. sinC + sinD = 2sin\left ( \frac{C+D}{2} \right )cos\left ( \frac{C-D}{2} \right )
  2. sinC – sinD = 2cos\left ( \frac{C+D}{2} \right )sin\left ( \frac{C-D}{2} \right )
  3. cosC + cosD = 2cos\left ( \frac{C+D}{2} \right )cos\left ( \frac{C-D}{2} \right )
  4. cosC – cosD = 2sin\left ( \frac{C+D}{2} \right )sin\left ( \frac{D-C}{2} \right )

 

Trigonometry Important Questions-SSC CGL-PART 1

Trigonometry Important Questions-SSC CGL-PART 2

 

 

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