Trigonometric Ratios of Compound Angles

Trigonometric Ratios of Compound Angles.

Let A and B two angles. Then their sum A + B or the difference A - B is called a compound angle

(a) Trigonometric ratios of A + B (Addition formula)

Let a revolving line start from OX and trace out an angle XOY = A and revolve further through an angle YOZ = B

∴ ∠XOZ = A + B

Let P be any point in OZ. Draw PM perpendicular to OX and PN perpendicular to OY From N draws NQ perpendicular to OX ad NR perpendicular to MP.

Here, ∠RPN = 90 - ∠PNR

trigonometric ratios of A+B (Addition formula)
trigonometric ratios of A+B (Addition formula)

= ∠RNO

= ∠NOQ

= A

Again RMQN is a rectangle, so, MR = QN and RN = MQ

Now, sin(A + B) =\(\frac{MP}{OP}\) = \(\frac{MR + RP}{OP}\) =\(\frac{QN + RP}{OP}\) = \(\frac{QN}{OP}\) + \(\frac{RP}{OP}\) = \(\frac{QN}{ON}\) \(\frac{ON}{OP}\) + \(\frac{RP}{NP}\) \(\frac{NP}{OP}\)

cos(A + B) = \(\frac{OM}{OP}\) = \(\frac{OQ - MQ}{OP}\) =\(\frac{OQ - RN}{OP}\) = \(\frac{OQ}{OP}\) - \(\frac{RN}{OP}\) = \(\frac{OQ}{ON}\) \(\frac{ON}{OP}\) - \(\frac{RN}{NP}\) \(\frac{NP}{OP}\)

cosA cosB = sinA sinB

Hence, sin formula of compound angle(A + B) is sin (A + B) = sinA cosB + cosA sinB and consine formula of compound angle (A + B) and cos(A + B) = cosA cosB - sinA sinB

(b) Trigonometric Ratios of A - B(Subtraction formula)

Let a revolving line start from OX and trace out an angle XOY = A and then revolve back through an angle YOZ = B

∴ ∠XOZ = A - B

Let P be any point in the Line OZ. Draw PM perpendicular to OX and PN perpendicular to OY.

From N Draw NQ perpendicular to OX and perpendicular to MP produced.

Here, ∠RPN = 900 - ∠PNR

=∠PNY

=∠XOY

= A

Again QMRN is a rectangle. So, QN = MR and QM = NR

Now sin(a-b) =\(\frac{PM}{OP}\) = \(\frac{MR - PR}{OP}\) =\(\frac{QN - PR}{OP}\)

=\(\frac{QN}{OP}\) - \(\frac{PR}{OP}\) =\(\frac{QN}{ON}\) \(\frac{ON}{OP}\) - \(\frac{PR}{NP}\) \(\frac{NP}{OP}\)

= sinA cosB - cosA sinB

cos(A - B) = \(\frac{OM}{OP}\) =\(\frac{OQ + QM}{OP}\) =\(\frac{OQ + NR}{OP}\) =\(\frac{OQ}{OP}\) + \(\frac{NR}{OP}\) =\(\frac{OQ}{ON}\) \(\frac{ON}{OP}\) - \(\frac{NR}{NP}\) \(\frac{NP}{OP}\)

=cosA cosB + sinA sinB

Hence sine formula of compound angle (A - B) is sin (A - B) = sinA cosB - cosA sinB and cosine formula of compound angle (A - B) is cos (A - B) = cosA cosB + sinA sinB

Alternative Method

Take a unit circle with centre at the origin. Let the circle intersect the X-axis at the point P. Then the coorinates of P are (1.0)

Let Q be another point on the circumference of the circle such that∠POQ = A. Then the coordinates of Q are (cosA, sinA).

Let R be another point on the cirumference of the circle such that ∠QOR = B

Then ∠POR = ∠POQ + ∠QOR = A + B

So coordinates of R are ( cos(A + B) , sin(A + B)).

Take a point S on the circumference such that ∠POS = -B.

Then coordinates of the points S are (cos(-B), sin(-B)) = (cosB, sin(-B))

Here, ∠SOQ = ∠SOP + ∠POQ = A + B and ∠POR = ∠POQ + ∠QOR = A + B

∴ ∠SOQ = ∠POR

So, arc QS = arc PR

∴ Chord QS = Chord PR.

Now by distance formula

PR2 = [cos(A + B)-1]2 + [sin(A + B) - 0]2

= cos2 (A + B) - 2cos(A + B) + 1 + sin2 (A + B) = 2 - 2cos (A + B)

QS2 =(cosA - cosB)2 + [sinA - sin(-B)]2 = (cosA - cosB)2 + (sinA + sinB)2

= cos2A - 2cosA.cosB + cos2B + sin2A + 2sinA.sinB + sin2B

= 2 - 2cosA.cosB + 2sinA.sinB

Now, PR2 = QS2

or, 2 - 2cos(A + B) = 2 - 2cosA.cosB + 2sinA.sinB

or, cos(A + B) = cosA.cosB - sinA.sinB ........(i)

If the angle B is replaced by (-B), Then

Cos(A-B) = cosA.cos(-B) - sinA.sin(-B) = cosA.cosB + sinA.sinB .........(ii)

Again, cos \begin{bmatrix} \frac {\pi}2 - (A + B)\\ \end{bmatrix} = cos \begin{bmatrix}( \frac{\pi}2 - A) - B\\ \end{bmatrix}

or, sin(A + B) = cos (\(\frac {\pi}{2}\) - A) cos B + sin (\(\frac{\pi}{2}\) - A\) sin B = sinA cosB + cosA sinB ....... (iii)

Similarly, cos \begin{bmatrix} \frac {\pi}2 - (A + B)\\ \end{bmatrix} = cos \begin{bmatrix}( \frac{\pi}2 - A) + B\\ \end{bmatrix}

or, sin(A - B) = cos ( \(\frac{\pi}{2}\) - A) cosB - sin( \(\frac{\pi}{2}\) - A) sinB = sinA cosB - cosA sinB .......... (iv)

(c) Tangent formula of compound angle (A + B)

tan (A + B) = \(\frac{sin(A + B)}{cos(A + B)}\) = \(\frac{sinA cosB + cosA sinB}{cosA cosB - sinA sinB}\) = \(\frac {\frac {sinA cosB}{cosA cosB} + \frac {cosA sinB}{cosA cosB}}{\frac {cosA cosB}{cosA cosB} - \frac {sinA sinB}{cosA cosB}}\)

= \(\frac{tan A + tan B}{1 - tanA tanB}\)

(d) Tangent formula of compound angle (A - B)

tan (A - B) = \(\frac{sin(A - B)}{cos(A - B)}\)

=\(\frac{sinA cosB - cosA sinB}{cosA cosB + sinA sinB}\)

=\(\frac{\frac{sinA cosB}{cosA cosB} - \frac{cosA sinB}{cosA cosB}}{\frac{cosA cosB}{cosA cosB} + \frac{sinA sinB}{cosA cosB}}\)

(e) Cotangent formula of compound angle (A + B)

cot (A + B) = \(\frac{cos(A + B)}{sin(A + B)}\)

=\(\frac{cosA cosB - sinA sinB}{sinA cosB + cosA sinB}\)

=\(\frac{\frac{cosA cosB}{sinA sinB} - \frac{sinA sinB}{sinA sinB}}{\frac{sinA cosB}{sinA sinB} + \frac{cosA sinB}{sinA sinB}}\)

=\(\frac{cotA cotB - 1}{cotB + cotA}\)

(f) Cotangent formula of compound angle (A - B)

cot(A - B) = \(\frac{cos(A - B)}{sin(A - B)}\)

=\(\frac{cosA cosB + sinA sinB}{sinA cosB - cosA sinB}\)

=\(\frac{\frac{cosA cosB}{sinA sinB} + \frac{sinA sinB}{sinA sinB}}{\frac{sinA cosB}{sinA sinB} - \frac{cosA sinB}{sinA sinB}}\)

Trigonometric Ratios of Compound Angles
sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB cos(A - B) = cosA cosB + sinA sinB
tan(A + B) = \(\frac{tan A + tan B}{1 - tanA tanB}\) tan(A - B) =\(\frac{tanA - tanB}{1 + tanA tanB}\)
cot(A + B) =\(\frac{cotA cotB - 1}{cotB + cotA}\) cot(A - B) =\(\frac{cotA cotB + 1}{cotB - cotA}\)

Some more results :

1. sin(A + B). sin(A - B) = cos2B - cos2A

Proof:

LHS = sin(A + B) .sin(A - B) = (sinA cosB + cosA sinB) . (sinA cosB - cosA sinB)

= sin2A cos2B - cos2A sin2B = (1 - cos2A) cos2B - cos2A(1 - cos2B)

= cos2B - cos2A cos2B - cos2A + cos2A cos2B = cos2B - cos2A

2. sin(A + B). sin(A - B) = sin2A - sin2B

proof:

L.H.S = sin(A + B). sin(A - B) = cos2B - cos2A = 1 - sin2B - (1 - sin2A) = sin2A - sin2B

3. cos(A + B). cos(A - B) = cos2A - sin2B

Proof :

LHS = cos (A + B). cos(A - B) = (cosA cosB - sinA sinB) (cosA cosB + sinA sinB)

= cos2A cos2B - sin2A sin2B = cos2A(1 - sin2B) - (1 - cos2A) sin2B

= cos2A - cos2A sin2B - sin2B + cos2A sin2B

= cos2A - cos2A sin2B - sin2B + cos2A sin2B

=cos2A - sin2A

4. cos(A + B) . cos(A - B) = cos2B - sin2A

Proof :

LHS = cos(A + B) . cos(A - B) = cos2A - sin2B = 1 - sin2A - (1 - cos2B) = cos2B - sin2A

5. cot(A + B) .cot(A - B) =\(\frac{cot^{2} A. cot^{2} B - 1}{cot^{2} B - cot^{2} A}\)

LHS = cot(A + B). cot(A - B) = ( \(\frac{cotA. cotB - 1}{cotB + cotA}\)). ( \(\frac{cotA .cotB + 1}{cotB - cotA}\)) =\(\frac{cot^{2}A . cot^{2}B}{cot^{2}B - cot^{2}A}\)

6. tan(A + B). tan(A - B) =\(\frac{tan^{2}A - tan^{2}B}{1 - tan^{2}A . tan^{2}B}\)

Proof :

LHS = tan(A + B). tan(A - B) =( \(\frac{tanA + tanB}{1 - tanA.tanB}\)) . \(\frac{tanA - tanB}{1 + tanA tanB}\) = \(\frac{tan^{2}A - tan^{2}B}{1 - tan^{2}A tan^{2}B}\)

7. sin(A + B + C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC - sinA sinB sinC

Proof :

LHS = sin(A + B + C) = sin(A + B) cosC + cos(A + B) sinC

= (sinA cosB + cosA sinB) cosC + (cosA cosB - sinA sinB) sinC

= sinA cosB cosC + cosA sinB cosC + cosA cosB sinC - sinA sinB sinC

8. cos(A + B + C) = cosA.cosB.cosC - cosA.sinB.sinC - sinC.cosB.sinA - sinA.sinB.cosC

Proof:

LHS = cos(A + B + C) = cos(A + B) cosC - sin(A + B) sinC

=(cosA cosB - sinA sinB) cosC - (sinA cosB + cosA sinB) sinC

= cosA.cosB.cosC - sinA sinB cosC - sinC.cosB.sinA - cosA.sinB.sinC

9. tan (A + B + C) =\(\frac{tanA + tanB + tanC - tanA tanB tanC}{1 - tanB tanC - tanC tanA - tanA tanB}\)

Proof:

LHS = tan(A + B + C) = \(\frac{tan(A + B) + tanC}{1 - tan (A + B) tanC}\)

= \(\frac{\frac{tanA + tanB}{1 - tanA tanB} + tanC}{1 -(\frac{tanA + tanB}{1 - tanA tanB}) tanC}\)

=\(\frac{tanA + tanB + tanC - tanA tanB tanC}{1 - tanB tanC - tanC tanA - tanA tanB}\)

Trigonometric Ratios of Compound Angles
sin(A + B) = sinA cosB + cosA sinB

 

sin(A - B) = sinA cosB - cosA sinB

 

cos(A + B) = cosA cosB - sinA sinB cos(A - B) = cosA cosB + sinA sinB
tan(A + B) = \(\frac{tan A + tan B}{1 - tanA tanB}\) tan(A - B) =\(\frac{tanA - tanB}{1 + tanA tanB}\)
cot(A + B) =\(\frac{cotA cotB - 1}{cotB + cotA}\) cot(A - B) =\(\frac{cotA cotB + 1}{cotB - cotA}\)

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  • Find the value without using a calculator .

    sin 15(^o)

    (frac{sqrt{3}-1}{2 sqrt{2} })


    90(^o)


    (frac{sqrt{3}-3}{3 sqrt{3} })


    (frac{sqrt{2}-1}{2 sqrt{3} })


  • Find the value without using a calculator .

    Sin 75(^o)

    (frac{sqrt{3}+1}{3sqrt{1}})


    (frac{sqrt{3}+3}{2sqrt{3}})


    (frac{sqrt{3}+1}{2sqrt{2}})


    (frac{sqrt{5}+2}{3sqrt{1}})


  • Find the value without using a calculator .

    Sin 105(^o)

    (frac{sqrt{2+1}}{3sqrt{1}})


    (frac{sqrt{3+1}}{2sqrt{2}})


    (frac{sqrt{3+1}}{3sqrt{4}})


    (frac{sqrt{3+1}}{4sqrt{2}})


  • Find the value without using a calculator .

    Sin 135(^o)

    (frac{2}{sqrt{4}})


    (frac{2}{sqrt{-2}})


    (frac{1}{sqrt{2}})


    (frac{2}{sqrt{1}})


  • Find the value without using calculator .

    Cos 105(^0)

    (frac{1-sqrt{2}}{2sqrt{2}})


    (frac{1-sqrt{3}}{2sqrt{2}})


    (frac{1-sqrt{2}}{2sqrt{3}})


    (frac{1-sqrt{1}}{1sqrt{1}})


  • Find the value without using calculator . 

    tan 15(^o)

    3 - (sqrt{-3})


    2 - (sqrt{-3})


    2 - (sqrt{3})


    -2 + (sqrt{-3})


  • Evaluate without using a calculator.

    tan 165 (^o)

    (sqrt{2}) - 1


    1.005423


    (sqrt{3}) - 2


    (sqrt{2}) - 3


  • tan 70(^o) = tan 20(^o) + 2 tan 50 (^o)

     1.009 , 12


    1 (frac{25}{24})


    1 (frac{24}{25})


     1 (frac{24}{24})


  • tan 50 (^o) = tan 40 (^o) + 2 tan 10 (^o)

     - (frac{119}{119}) , (frac{120}{119})


     - (frac{112}{162}) , (frac{122}{129})


    tan 60


     - (frac{119}{169}) , (frac{120}{169})


  • tan 65 (^0) - tan 25 (^0) = 2 tan 40 (^o)

    0.554


     - (frac{119}{169}) , (frac{120}{169})


    11


    1


  • If cot α = (frac{1}{2}) and sec β = (frac{5}{3}) , find tan ( α + β).

    -1.009


    3


    -1.224


    (frac{1}{2})


  • If sin A = (frac{3}{5}) & sin B = (frac{12}{13}) , find the value of cos (A - B).

    (frac{66}{65})


    (frac{54}{65})


    (frac{56}{65})


    (frac{55}{55})


  • If cosα = (frac{4}{5}) and cosβ = (frac{12}{13}) then find the value of cos (α+β).

     

    (frac{36}{35})


    (frac{33}{35})


    1.090


    (frac{33}{65})


  • If sin A = (frac{3}{5}) and cos B = (frac{15}{17}) then find the value of sin (A + B).

    (frac{80}{84})


    (frac{85}{85})


    (frac{77}{85})


    (frac{77}{77})


  • If tan A = (frac{3}{4}) and tan B = (frac{5}{12}) then find the value of sin (A - B).

    (frac{65}{15})


    (frac{12}{60})


    (frac{10}{60})


    (frac{15}{65})


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sasmeeta bhandari

2tan50 tan20 is equals to cot20