# Trigonometry

Trigonometrical ratios

SinA = $\frac{{\rm{p}}}{{\rm{h}}}$, cosA = $\frac{{\rm{b}}}{{\rm{h}}}$, tanA = $\frac{{\rm{p}}}{{\rm{b}}}$, cotA = $\frac{{\rm{b}}}{{\rm{p}}}$, SecA = $\frac{{\rm{h}}}{{\rm{b}}}$ and cosecA = $\frac{{\rm{h}}}{{\rm{p}}}.$

Where, p = perpendicular, b = base and h = hypotenuse and h2 = p2 + b2.

2. SinA * cosecA = 1, cosA * secA = 1, tanA * cotA = 1.

tanA = $\frac{{{\rm{sinA}}}}{{{\rm{cosA}}}}$, cotA = $\frac{{{\rm{cosA}}}}{{{\rm{sinA}}}}$

3. sin2A + cos2A = 1, sec2A – tan2A = 1, cosec2A – cot2A = 1

4. The trigonometric values of different degree(0° - 180°) are listed below:

 Angles Ratios Sin Cos Tan Cosec Sec Cot 00 0 1 0 1 300 $\frac{1}{2}$ $\frac{{\sqrt 3 }}{2}$ $\frac{1}{{\sqrt 3 }}$ 2 $\frac{2}{{\sqrt 3 }}$ $\sqrt 3$ 450 $\frac{1}{{\sqrt 2 }}$ $\frac{1}{{\sqrt 2 }}$ 1 $\sqrt 2$ $\sqrt 2$ 1 600 $\frac{{\sqrt 3 }}{2}$ $\frac{1}{2}$ $\sqrt 3$ $\frac{2}{{\sqrt 3 }}$ 2 $\frac{1}{{\sqrt 3 }}$ 900 1 0 1 0 1200 $\frac{{\sqrt 3 }}{2}$ $- \frac{1}{2}$ $- \sqrt 3$ $\frac{2}{{\sqrt 3 }}$ $- 2{\rm{ }}$ $- \frac{1}{{\sqrt 3 }}{\rm{ }}$ 1350 $\frac{1}{{\sqrt 2 }}$ $- \frac{1}{{\sqrt 2 }}$ -1 $\sqrt 2$ $- \sqrt 2$ -1 1500 $\frac{1}{2}$ $- \frac{{\sqrt 3 }}{2}$ $- \frac{1}{{\sqrt 3 }}$ 2 $- \frac{2}{{\sqrt 3 }}$ $- \sqrt 3$ 1800 0 -1 0 -1

5.With angle is (-A),

Sin(-A) = -sinA, cos(-A) = cosA. Tan(-A) = - tanA, etc.

6. When angle is (90° - A) and (90° + A)

Sin(90° - A) = cosA sin(90°+A) = cosA

Cos(90° - A) = sinA cos(90° + A) = -sinA

Tan(90° - A) = cotA tan(90°+A) = -cotA etc.

7. With angle (180° - A) and (180° + A)

Sin(180° - A) = sinA sin(180°+A) = -sinA

Cos(180° - A) = -cosA cos(180° + A) = -cosA

Tan(180° - A) = -tanA tan(180°+A) = tanA etc.

8. When angle is (270° - A) and (270° + A)

Sin(270° - A) = -cosA sin(270°+A) = -cosA

Cos(270° - A) = -sinA cos(270° + A) = sinA

Tan(270° - A) = cotA tan(270°+A) = -cotA etc.

9. With angle (360° - A) and (360° + A)

Sin(360° - A) = -sinA sin(360°+A) = sinA

Cos(360° - A) = cosA cos(360° + A) = cosA

Tan(360° - A) = -tanA tan(360°+A) = tanA etc.

Trigonometric Ratios of compound angles:

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

2. cos(A + B) = cosA.cosB – sinA.sinB

3. Tan (A + B) = $\frac{{{\rm{tanA}} + {\rm{tanB}}}}{{1 - {\rm{tanA}}.{\rm{tanB}}}}$

4. sin (A – B) = sinA.cosB - cosA.sinB

5. cos(A – B) = cosA.cosB + sinA.sinB

6. tan(A – B) = $\frac{{{\rm{tanA}} - {\rm{tanB}}}}{{1 + {\rm{tanA}}.{\rm{tanB}}}}$

Trigonometric ratios of multiple angles:

1.

(i)sin2A = 2sinA.cosA

(ii)cos2A = cosA2A – sin2A = 2cos2A – 1 = 1 – 2sin2A

(iii)tan2A = $\frac{{2{\rm{tanA}}}}{{1 - {{\tan }^2}{\rm{A}}}}$

(iv)sin2A = $\frac{{2{\rm{tanA}}}}{{1 + {{\tan }^2}{\rm{ A}}}}$

(v)cos2A = $\frac{{1 - {{\tan }^2}{\rm{A}}}}{{1 + {{\tan }^2}{\rm{A}}}}$

2.

(i) sin3A = 3sinA – 4sin3A

(ii) cos3A = 4cos3A – 3cosA

(iii) tan3A = $\frac{{3{\rm{tanA}} - {{\tan }^3}{\rm{A}}}}{{1 - 3{{\tan }^2}{\rm{A}}}}{\rm{ }}$

Trigonometric ratios of Sub-multiple angles:

1.

(i)sinA = 2sin$\frac{{\rm{A}}}{2}$.cos$\frac{{\rm{A}}}{2}$.

(ii)cosA = cosA2$\frac{{\rm{A}}}{2}$ – sin2$\frac{{\rm{A}}}{2}$. = 2cos2$\frac{{\rm{A}}}{2}$. – 1 = 1 – 2sin2$\frac{{\rm{A}}}{2}$.

(iii)tanA = $\frac{{2{\rm{tan}}\frac{{\rm{A}}}{2}.}}{{1 - {{\tan }^2}\frac{{\rm{A}}}{2}.}}$

(iv) sinA = 3sin$\frac{{\rm{A}}}{3}$ – 4sin3$\frac{{\rm{A}}}{3}$

(v) cosA = 4cos3$\frac{{\rm{A}}}{3}$ – 3cos$\frac{{\rm{A}}}{3}$

(vi) tanA = $\frac{{3{\rm{tan}}\frac{{\rm{A}}}{3} - {{\tan }^3}\frac{{\rm{A}}}{3}}}{{1 - 3{{\tan }^2}\frac{{\rm{A}}}{3}}}$

Transformation of Trigonometric Formulae:

1.2sinA.cosB = sin(A + B) + sin(A – B)

2. 2cosA.sinB = sin(A+B) – sin(A – B)

3. 2cosA.cosB = cos(A + B) + cos(A – B)

4. 2sinA.sinB = cos(A + B) – cos(A + B)

5. sinC + sinD = 2sin $\frac{{{\rm{C}} + {\rm{D}}}}{2}.$ cos $\frac{{{\rm{C}} - {\rm{D}}}}{2}$

6. sinC – sinD = 2cos $\frac{{{\rm{C}} + {\rm{D}}}}{2}.$ sin $\frac{{{\rm{C}} - {\rm{D}}}}{2}$

7. cosC + cosD = 2cos $\frac{{{\rm{C}} + {\rm{D}}}}{2}$. cos $\frac{{{\rm{C}} - {\rm{D}}}}{2}$

8. cosC – cosD = -2sin${\rm{ }}\frac{{{\rm{C}} + {\rm{D}}}}{2}$.sin $\frac{{{\rm{C}} - {\rm{D}}}}{2}$

Where, A = $\frac{{{\rm{C}} + {\rm{D}}}}{2}$ and B = $\frac{{{\rm{C}} - {\rm{D}}}}{2}$

Conditional Trigonometric Identities:

If A + B + C = π

1.sinA = sin[ π– (B + C)] = sin(B + C)

2. cosA = cos[π – (B + C)] = -cos(B + C)

3. tanA = tan[π – (B + C)] = -tan(B + C)etc.

If, $\frac{{\rm{A}}}{2}{\rm{ }}$+ $\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$ = $\frac{{\rm{\pi }}}{2}$

1.sin$\frac{{\rm{A}}}{2}{\rm{ }}$= sin[$\frac{{\rm{\pi }}}{2}$ – ($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$)] = cos($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$)

2. cos$\frac{{\rm{A}}}{2}{\rm{ }}$ = cos[$\frac{{\rm{\pi }}}{2}$ – ($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$))] = sin($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$))

3. tan$\frac{{\rm{A}}}{2}{\rm{ }}$ = tan[$\frac{{\rm{\pi }}}{2}$ – ($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$))] = cot($\frac{{\rm{B}}}{2}$ + $\frac{{\rm{C}}}{2}$))etc.

Example 1

Prove:

cos2(A – 120°) + cos2A + cos2(A – 120°) = $\frac{3}{2}$

L.H.S. = cos2(A – 120°) + cos2A + cos2(A – 120°)

= {cos2(A – 120°)}2 + cos2 A + {cos2(A + 120°)}2

= (cosA.cos120°+sinA.sin120°) + cos2A + (cosA.cos120°– sinA.sin120°)

= ${\left( { - \frac{1}{2}{\rm{\: cosA}} + \frac{{\sqrt 3 }}{2}{\rm{sinA}}} \right)^2}$+ cos2A + ${\left( { - \frac{1}{2}{\rm{\: cosA}} + \frac{{\sqrt 3 }}{2}{\rm{sinA}}} \right)^2}$

= $\frac{1}{4}$cos2A – 2.$\frac{1}{2}$.$\frac{{\sqrt 3 }}{2}$ cosA.sinA + $\frac{3}{4}{\rm{\: }}$sin2A + cos2A + $\frac{1}{4}{\rm{\: }}$cos2A + 2.$\frac{1}{2}$.$\frac{{\sqrt 3 }}{2}$ cosA.sinA +$\frac{3}{4}{\rm{\: }}$sin2A

= $\frac{1}{4}$cos2 A + $\frac{3}{4}$ sin2A + cos2A + $\frac{1}{4}$ cos2A + $\frac{3}{4}$ sin2A

= $\frac{2}{4}$ cos2A + $\frac{6}{4}$sin2A + cos2A.

= $\frac{1}{4}$ (2cos2A + 6sin2A + 4cos2A) = $\frac{1}{4}$(6cos2A + 6sin2A)

= $\frac{6}{4}$(cos2A + sin2A) = $\frac{3}{2}$ = R.H.S.

Example 2

If A + B+ C = π prove that cos2A + cos2B + cos2C = 1 – 2cosA.cosB.cosC

Soln:

L.H.S. =cos2A + cos2B + cos2C

= $\frac{{1 + {\rm{cos}}2{\rm{A}}}}{2}$ + $\frac{{1 + {\rm{cos}}2{\rm{B}}}}{2}$ + cos2C.

= $\frac{1}{2} + \frac{1}{2} + \frac{1}{2}$ (cos2A + cos2B) + cos2C.

= 1 + $\frac{1}{2}$. 2 cos $\frac{{2{\rm{A}} + 2{\rm{B}}}}{2}$. Cos $\frac{{2{\rm{A}} - 2{\rm{B}}}}{2}$ + cos2C.

= 1 + cos(A + B).cos(A – B) + cos2C.

= 1 – cosC.cos(A – B) + cos2C[A + B = π – C, So, cos(A+B) = cos(π – C) = – cosC]

= 1 – cosC {cos(A – B) – cosC}

= 1 – cosC {cos(A – B) + cos(A + B)}

= 1 – cosC.(cosA.cosB + sinA.sinB + cosA.cosB – sinA.sinB)

= 1 – cosC.(2cosA.cosB) = 1 – 2cosA.cosB.cosC = R.H.S.

Example 3

Solve:

2 sin2x – sinx = 0

Soln:

Here, 2 sin2x = sinx

Or, 4sinx cosx – sinx = 0

Or, sinx (4cosx – 1) = 0

Either, sinx = 0

So, x = nπ

Or, 4cosx – 1 = 0

Or, cosx = 1414

Or, cosx = cosα [cosx = 1414]

So, x = 2nπ ±± α.

So, x = 2nπ ±± cos–11414 n ԑ Z.

Height and Distance

Angle of elevation: The angle of elevation is defined as angle between the line of sight and horizontal line made by the observer when the observer observes the object above the horizontal line.

Example 4

A man observes the top of a pole 52cm height situated in front of him and finds the angle of elevation to be 30° .If the distance between man and pole is 86m .Find the height of that man.

Soln

Tan30° =$\frac{{{\rm{ED}}}}{{{\rm{AD}}}}$=$\frac{{{\rm{ED}}}}{{86}}$

$\frac{1}{{\sqrt 3 }}$=$\frac{{{\rm{ED}}}}{{86}}$

Or, ED = $\frac{{86}}{{\sqrt 3 }}$

Or, ED = 49.65cm

Height of man (DC) = EC – ED = 50- 47.65 =2.35 m

Example 5

The angle of elevation from the roof of a house to the top of atop of tree is found to be 30°. If the height of the house and tree are 8 m and 20m respectively, find the distance between the house and the tree.

Soln

ED = EC - DC = 20- 8 = 12 cm

Tan 30°=$\frac{{{\rm{ED}}}}{{{\rm{AD}}}}$

$\frac{1}{{\sqrt 3 }}$=$\frac{{12}}{{{\rm{AD}}}}$