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Popular Trigonometry >

cos(216)=cos^2(a)-sin^2(a)

Solution

cos (21 6^{\circ}) = cos^{2} (a) - sin^{2} (a)

Solution

a = 1.25663 \dots + 36 0^{\circ} n, a = 18 0^{\circ} - 1.25663 \dots + 36 0^{\circ} n, a = - 1.25663 \dots + 36 0^{\circ} n, a = 18 0^{\circ} + 1.25663 \dots + 36 0^{\circ} n

Radians

a = 1.25663 \dots + 2 πn, a = π - 1.25663 \dots + 2 πn, a = - 1.25663 \dots + 2 πn, a = π + 1.25663 \dots + 2 πn

Solution steps

cos (21 6^{\circ}) = cos^{2} (a) - sin^{2} (a)

cos (21 6^{\circ}) = - \frac{5 + 1}{4}

cos (21 6^{\circ})

Rewrite using trig identities:

cos (14 4^{\circ})

cos (21 6^{\circ})

Use the following identity:

cos (x) = cos (36 0^{\circ} - x)

cos (x)

Use the following property:

cos (θ) = cos (- θ)

cos (x) = cos (- x)

= cos (- x)

Apply the periodicity of

cos

cos (36 0^{\circ} + θ) = cos (θ)

cos (- x) = cos (36 0^{\circ} - x)

= cos (36 0^{\circ} - x)

= cos (36 0^{\circ} - 21 6^{\circ})

Simplify:

36 0^{\circ} - 21 6^{\circ} = 14 4^{\circ}

36 0^{\circ} - 21 6^{\circ}

Convert element to fraction:

36 0^{\circ} = 36 0^{\circ}

= 36 0^{\circ} - 21 6^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{36 0 ^{\circ} 5 - 108 0 ^{\circ}}{5}

36 0^{\circ} 5 - 108 0^{\circ} = 72 0^{\circ}

36 0^{\circ} 5 - 108 0^{\circ}

Multiply the numbers:

2 \cdot 5 = 10

= 180 0^{\circ} - 108 0^{\circ}

Add similar elements:

180 0^{\circ} - 108 0^{\circ} = 72 0^{\circ}

= 72 0^{\circ}

= 14 4^{\circ}

= cos (14 4^{\circ})

= cos (14 4^{\circ})

Rewrite using trig identities:

- cos (3 6^{\circ})

cos (14 4^{\circ})

Use the basic trigonometric identity:

cos (x) = - cos (18 0^{\circ} - x)

= - cos (18 0^{\circ} - 14 4^{\circ})

Simplify:

18 0^{\circ} - 14 4^{\circ} = 3 6^{\circ}

18 0^{\circ} - 14 4^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}

= 18 0^{\circ} - 14 4^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{18 0 ^{\circ} 5 - 72 0 ^{\circ}}{5}

Add similar elements:

90 0^{\circ} - 72 0^{\circ} = 18 0^{\circ}

= 3 6^{\circ}

= - cos (3 6^{\circ})

= - cos (3 6^{\circ})

Rewrite using trig identities:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= - \frac{5 + 1}{4}

- \frac{5 + 1}{4} = cos^{2} (a) - sin^{2} (a)

Subtract

cos^{2} (a) - sin^{2} (a)

from both sides

- \frac{5 + 1}{4} - cos^{2} (a) + sin^{2} (a) = 0

Simplify

- \frac{5 + 1}{4} - cos^{2} (a) + sin^{2} (a) : \frac{- 5 - 1 - 4 cos ^{2} ( a ) + 4 sin ^{2} ( a )}{4}

- \frac{5 + 1}{4} - cos^{2} (a) + sin^{2} (a)

Convert element to fraction:

cos^{2} (a) = \frac{cos ^{2} ( a ) 4}{4}, sin^{2} (a) = \frac{sin ^{2} ( a ) 4}{4}

= - \frac{5 + 1}{4} - \frac{cos ^{2} ( a ) \cdot 4}{4} + \frac{sin ^{2} ( a ) \cdot 4}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- ( 5 + 1 ) - cos ^{2} ( a ) \cdot 4 + sin ^{2} ( a ) \cdot 4}{4}

Expand

- (5 + 1) - cos^{2} (a) \cdot 4 + sin^{2} (a) \cdot 4 : - 5 - 1 - cos^{2} (a) \cdot 4 + sin^{2} (a) \cdot 4

- (5 + 1) - cos^{2} (a) \cdot 4 + sin^{2} (a) \cdot 4

= - (5 + 1) - 4 cos^{2} (a) + 4 sin^{2} (a)

- (5 + 1) : - 5 - 1

- (5 + 1)

Distribute parentheses

= - (5) - (1)

Apply minus-plus rules

+ (- a) = - a

= - 5 - 1

= - 5 - 1 - cos^{2} (a) \cdot 4 + sin^{2} (a) \cdot 4

= \frac{- 5 - 1 - 4 cos ^{2} ( a ) + 4 sin ^{2} ( a )}{4}

\frac{- 5 - 1 - 4 cos ^{2} ( a ) + 4 sin ^{2} ( a )}{4} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 5 - 1 - 4 cos^{2} (a) + 4 sin^{2} (a) = 0

Rewrite using trig identities

- 1 - 5 - 4 cos^{2} (a) + 4 sin^{2} (a)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 1 - 5 - 4 (1 - sin^{2} (a)) + 4 sin^{2} (a)

Simplify

- 1 - 5 - 4 (1 - sin^{2} (a)) + 4 sin^{2} (a) : 8 sin^{2} (a) - 5 - 5

- 1 - 5 - 4 (1 - sin^{2} (a)) + 4 sin^{2} (a)

Expand

- 4 (1 - sin^{2} (a)) : - 4 + 4 sin^{2} (a)

- 4 (1 - sin^{2} (a))

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = 1, c = sin^{2} (a)

= - 4 \cdot 1 - (- 4) sin^{2} (a)

Apply minus-plus rules

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (a)

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 sin^{2} (a)

= - 1 - 5 - 4 + 4 sin^{2} (a) + 4 sin^{2} (a)

Simplify

- 1 - 5 - 4 + 4 sin^{2} (a) + 4 sin^{2} (a) : 8 sin^{2} (a) - 5 - 5

- 1 - 5 - 4 + 4 sin^{2} (a) + 4 sin^{2} (a)

Add similar elements:

4 sin^{2} (a) + 4 sin^{2} (a) = 8 sin^{2} (a)

= - 1 - 5 - 4 + 8 sin^{2} (a)

Group like terms

= 8 sin^{2} (a) - 1 - 5 - 4

Subtract the numbers:

- 1 - 4 = - 5

= 8 sin^{2} (a) - 5 - 5

= 8 sin^{2} (a) - 5 - 5

= 8 sin^{2} (a) - 5 - 5

- 5 - 5 + 8 sin^{2} (a) = 0

Solve by substitution

- 5 - 5 + 8 sin^{2} (a) = 0

Let:

sin (a) = u

- 5 - 5 + 8 u^{2} = 0

- 5 - 5 + 8 u^{2} = 0 : u = \frac{2 5 + 5}{4}, u = - \frac{2 5 + 5}{4}

- 5 - 5 + 8 u^{2} = 0

Move

5

to the right side

- 5 - 5 + 8 u^{2} = 0

Add

5

to both sides

- 5 - 5 + 8 u^{2} + 5 = 0 + 5

Simplify

- 5 + 8 u^{2} = 5

- 5 + 8 u^{2} = 5

Move

5

to the right side

- 5 + 8 u^{2} = 5

Add

5

to both sides

- 5 + 8 u^{2} + 5 = 5 + 5

Simplify

8 u^{2} = 5 + 5

8 u^{2} = 5 + 5

Divide both sides by

8

8 u^{2} = 5 + 5

Divide both sides by

8

\frac{8 u ^{2}}{8} = \frac{5}{8} + \frac{5}{8}

Simplify

\frac{8 u ^{2}}{8} = \frac{5}{8} + \frac{5}{8}

Simplify

\frac{8 u ^{2}}{8} : u^{2}

\frac{8 u ^{2}}{8}

Divide the numbers:

\frac{8}{8} = 1

= u^{2}

Simplify

\frac{5}{8} + \frac{5}{8} : \frac{5 + 5}{8}

\frac{5}{8} + \frac{5}{8}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{5 + 5}{8}

u^{2} = \frac{5 + 5}{8}

u^{2} = \frac{5 + 5}{8}

u^{2} = \frac{5 + 5}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{5 + 5}{8}, u = - \frac{5 + 5}{8}

\frac{5 + 5}{8} = \frac{2 5 + 5}{4}

\frac{5 + 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 + 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{5 + 5}{2 2}

Rationalize

\frac{5 + 5}{2 2} : \frac{2 5 + 5}{4}

\frac{5 + 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 + 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

- \frac{5 + 5}{8} = - \frac{2 5 + 5}{4}

- \frac{5 + 5}{8}

Simplify

\frac{5 + 5}{8} : \frac{5 + 5}{2 2}

\frac{5 + 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 + 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{5 + 5}{2 2}

= - \frac{5 + 5}{2 2}

Rationalize

- \frac{5 + 5}{2 2} : - \frac{2 5 + 5}{4}

- \frac{5 + 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{5 + 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 5 + 5}{4}

= - \frac{2 5 + 5}{4}

u = \frac{2 5 + 5}{4}, u = - \frac{2 5 + 5}{4}

Substitute back

u = sin (a)

sin (a) = \frac{2 5 + 5}{4}, sin (a) = - \frac{2 5 + 5}{4}

sin (a) = \frac{2 5 + 5}{4}, sin (a) = - \frac{2 5 + 5}{4}

sin (a) = \frac{2 5 + 5}{4} : a = arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

sin (a) = \frac{2 5 + 5}{4}

Apply trig inverse properties

sin (a) = \frac{2 5 + 5}{4}

General solutions for

sin (a) = \frac{2 5 + 5}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

a = arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

a = arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

sin (a) = - \frac{2 5 + 5}{4} : a = arcsin (- \frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} + arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

sin (a) = - \frac{2 5 + 5}{4}

Apply trig inverse properties

sin (a) = - \frac{2 5 + 5}{4}

General solutions for

sin (a) = - \frac{2 5 + 5}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 36 0^{\circ} n, x = 18 0^{\circ} + arcsin (a) + 36 0^{\circ} n

a = arcsin (- \frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} + arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

a = arcsin (- \frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} + arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

Combine all the solutions

a = arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} - arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = arcsin (- \frac{2 5 + 5}{4}) + 36 0^{\circ} n, a = 18 0^{\circ} + arcsin (\frac{2 5 + 5}{4}) + 36 0^{\circ} n

Show solutions in decimal form

a = 1.25663 \dots + 36 0^{\circ} n, a = 18 0^{\circ} - 1.25663 \dots + 36 0^{\circ} n, a = - 1.25663 \dots + 36 0^{\circ} n, a = 18 0^{\circ} + 1.25663 \dots + 36 0^{\circ} n

Graph

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