Basics Trigonometry

other essential relations

sin²α + cos²α = 1

cos²α – sin²α = cos2α

2 + sin²α – 2cos²α = 3sin²α

sinα * cosα = [sin2α]/2

3sinα – 4 sin³α = 3sinα

4cos³α – 3cosα = cos3α

2tanα/(1 – tan²α) = tan2α

(3tanα -tan³α)/(1 – tan²α) = tan3α

(1 – cosα)/2 = sin²(α/2)

(1 + cosα)/2 = cos²(α/2)

(1 – cosα)/(1 + cosα) = tan²(α/2)

(1 – cosα)/sinα = sinα/(1 + cosα) = tan(α/2)

tanα * cotα = 1

1/cos²α = 1 + tan²α

1 /sin²α = 1 + cot²α

 

Addition theorems

sin(90° – α) = sin(90° + α) = cos(360° – α) = cos(-α) = cosα

cos(180° –  α) = cos(180° + α) = -cosα

sin(180° – α) = cos(90° – α) = sinα

sin(180° + α) = sin(360° – α) = sin(-α) = -sinα

tan(180° – α) = cot(90 + α) = tan(-α) = -tanα

tan(90 + α) = cot(180 – α) = cot(-α) = -cotα

sin(α +/- β) = sinα*cosβ +/- cosα*sinβ

cos(α +/- β) = cosα*cosβ -/+ sinα*sinβ

tan(α +/- β) = (tanα +/- tanβ)/(1 -/+ tanα*tanβ)

sinα + sinβ = 2sin[(α + β)/2]*cos[(α – β)/2)]

sinα – sinβ = 2cos[(α + β)/2]*sin[(α – β)/2]

cosα + cosβ = 2cos[(α + β)/2]*cos[(α – β)/2]

cosα – cosβ = -2sin[(α + β)/2]*sin[(α – β)/2]

 

Substitution

sinα, cosα, tanα and cotα can also be represented in substituted form.

we replace

tan(α/2) = u

Relations mentioned above are as follows:

(1 – cosα)/(1 + cosα) = tan²(α/2) = u²

and

(1 – cosα)/sinα = sinα/(1 + cosα) = tan(α/2) = u

we solve

(1 – cosα)/(1 + cosα) =  u²

after cosα and get:

1 – cosα = u² + u²cosα

u² + u²cosα + cosα = 1

cosα(u² + 1) = 1 – u²

cosα = (1 – u²)/(1 + u²)

now cosα can be replaced by what we’ve get now:

(1 – cosα)/sinα = tan(α/2) = u by (1 – u²)/(1 + u²)

we solve after sinα

[1- (1 – u²)/(1 + u²)]/sinα = u

multiply with sinα, we get:

[1- (1- u²)/(1 + u²)] = u*sinα

transform right into a fraction with common denominator:

[(1 + u²) – (1 – u²)]/(1 + u²) = u*sinα:

Counter left: Leave brackets out and drop them out

(take care of minus sign -,-u² = +u²!)

2u²/(1 + u²) = u*sinα

links und rechts durch u dividieren und rechts durch u kürzen:

Divide left and right by u and shorten by u, we get:

2u/(1 + u²) = sinα

or

sinα = 2u/(1 + u²)

tanα as we know, equals to

sinα/cosα

Both fractions are being divided:

2u/(1 + u²) / (1 – u²)/(1 + u²) = sinα/cosα = tanα

what we get, is (see first chapter sings, symbols, relationships)

2u(1 + u²)/(1 + u²)(1 – u²) = tanα

shorten by (1 + u²): and there we are:

tanα = 2u/(1 – u²)

finally the term of cotα

cotα = cosα/sinα = 1/tanα = (1 – u²)/2u

Summarized:

sinα = 2u/(1 + u²)

cosα = (1 – u²)/(1 + u²)

tanα = 2u/(1 – u²)

cotα = (1 – u²)/2u

 

More relationships in geometry

 

A = α, B = β, C = γ

Sinus rate
a / sinα = b / sinβ = c / sinγ

Cosine rate
For the right-angled triangle, the calculation of the edge length c Pytagoras applies:
a² + b² = c²
If angle C = γ > 0, <180 and ≠ 90 °, then c is calculated with the consine rate:
a² + b² – 2abcosγ = c²

Trigonometry finds wide application in mathematics, physics and technical science.

Other essential relations for the right-angled triangle

a = csinα

b = ccosα

a = btanα

b = acotα

 

sinα = a/c = cosβ

cosα = b/c = sinβ

tanα = a/b = cotβ

cotα = b/a = tanβ

 

Useful reading:

Formulas and Charts, Mathematics Physics, 3rd edition 1984, Orell Füssli Publishing Zurich (a yellow book)

---