Trigonometry

Trigonometry is crucial for astronomers as it allows them to calculate distances to celestial objects, determine their sizes, and understand their movements. By applying trigonometric principles to angles and distances observed from different points on Earth, astronomers can create accurate models of the universe. Additionally, trigonometry aids in the analysis of light and other forms of electromagnetic radiation emitted by stars and galaxies, enhancing our understanding of their properties and behaviors.

Trigonometry plays a crucial role in criminology, particularly in crime scene investigations and forensic analysis. It is used to calculate distances, angles, and trajectories, helping investigators reconstruct events and determine the positions of victims, suspects, and evidence. Accurate measurements derived from trigonometric principles can enhance the validity of evidence presented in court, thereby supporting the investigative process. Overall, it aids in creating a clearer understanding of spatial relationships in criminal cases.

The value of tan 75 degrees can be calculated using the angle sum identity for tangent: tan(75°) = tan(45° + 30°) = (tan 45° + tan 30°) / (1 - tan 45° * tan 30°). Since tan 45° = 1 and tan 30° = 1/√3, substituting these values gives tan 75° = (1 + 1/√3) / (1 - 1/√3) = (√3 + 1) / (√3 - 1). Simplifying this expression results in tan 75° = 2 + √3.

Sin + Cos Does NOT equal '1'.

However,

Sin^(2) + Cos^(2) = 1 = ( 1^(2)

It is by Pythagoras.

Remember in a right angled triangle

H^(2) = o^(2) + a^(2)

Assume ' h = 1'

Then 1^(2) = 1 = o^(2) + a^)2)

But o = hSin & a = hCos

Substituting

1^(2) = (hSin)^(2) + (hCos(^(2))

However, we assume h = 1

Hence

1^(2) = Sin^(2) + Cos^(2)

This is the Trig. Identity .

Sin( A + B) = SinACosB + CosASinB.

Hence

Sin55Cos35 + Cos555Sin35 =

Sin(55 + 35) = Sin(90) = 1

Verification

If you take the various Sin & Cos values and multiply then add, they will answer to '1'.

Hence./

(0.819153..)(0.819152...) +(0.573576...)(0,573586...) =

0.67100999 + 0.328989 = 0.99995... ~ 1. as required.

The equation that satisfies the condition "what divided by cosine squared theta equals one" is simply the expression itself. If we let ( x ) be the quantity, then the equation can be expressed as ( \frac{x}{\cos^2 \theta} = 1 ). Solving for ( x ) gives ( x = \cos^2 \theta ). Thus, ( \cos^2 \theta ) divided by ( \cos^2 \theta ) equals one.

Using Similar Triangles or Ratios.

348:6 :: x:2

Form a fractional equation.

348/6 = x/2

X = 2(348)/6

x = 348/3

x = 116 m

NB THe colons (:) are mathematical shorthand for 'as to'. Used in ratios.

Use the Tangent function

Tan(angle) = opposite(height) / adjacent(shadow)

Substituting

Tan )53) = height/ 12m

Algebraically rearrange

height = 12m X Tan (53)

NB Make sure your calculator is in 'Degree' Mode.

Then type in '12' 'X' , '(' , 'Tan', '53', ')' , '=', The answer should 'pop ip' pm the screen .

height = 15.92453786...m

Approx. ht. ~ 15.92 m ( 2 d.p.).

Ue tangent trig. function

Tan(angle) = opposite(height) / Adjacent(base)

Tan (angle) = 40ft/58ft.

Cancel fraction down by '2'

Tan (angle) = 20/29

Convert fraction to a decimal , by dividing '29' into '20;'.

Tan(angle) = 0.689655172.....

Make sure your calculator is in 'Degree'(D) Mode . NEITHER Radian mode, NOR 'G' mode. Use button 'DRG' as required.

Then type in 'inverse function/second function/shift' , 'Tan' , The decimal above , ' =' , and the answer should 'pop up' on the screen.

Angle = ArcTan or Tan^(-1) ( 0.689655172....)

Angle = 34.59228860.... degrees. The answer!!!!!

'Delta; in maths means _'d_ifference' or 'change'.

NB Cube Root can be written as the exponent '1/3'

Hence

[x^(5)]^(1/3) =

x^(5/3)

r^(4) / r^(6) = r^(4-6) = r^(-2) = 1/r^(2)

Rules for manipulation of indicies.

#1 ; the coefficient MUST be the same . 'r' in this case

#2 ; for multiplication ; ; add the indices

#3 ; for division ; subtract the indices

#4 ' for 'nesting' ; multiply the indices

Careful something of the nature 'r^(2)' X s^(3)' CANNOT be done as the coefficients ''r' & 's' are different. The coefficients MUST be the same

Remember use the Pythagorean Trig/ Identity.

Sin^(2)(Theta) + Cos^(2)(Theta) = 1

Algebraically rearrange

Sin^(2)(Theta) = 1 - Cos^(2)(Theta)

Substitute

Sin^(2)(Theta) = 1 - 0.65^(2)

Factor

Sin^(2)(Theta) = ( 1- 0.65 )( 1 + 0.65)

Sin^(2)(Theta) = (0.35)(1.65)

Sin^(2)(Theta) = 0.5775

Sin(Theta) = sqrt(0.5775)

Sin(Theta) = 0.759934207....

Theta = Sun^(-1)(0.759934207...)

Theta = 49.45839813 degrees.

Pythagoras was a Classical Greek mathematician , who intorduced his famous eq;'n to western civilisation. .

He lived in about 500 BC on the island of Samos ( now in modern Greece).

Her a pre=Socratean philosopher and mathematician .

For right angled triangle with known sides of length and width we can find the hypotenuse (h) by applying the Pythagorean eq'n.

h^(2) = l^(2) + w^(2)

To find the angles we can use trigonometry(trig)

Sin(W) = length/hypotenuse

Sin(W) = l/h => W(degrees) = Sin^(-1)[l/h]

Cos(W) = w/h

Tan(W) = l/w

Complimentarily

Sin(L) = w/h

Cos(L) = l/h

Tan(L) = w/l

'L' & 'W' are the angles opposite to the given sides.

Area of as triangle = 0.5 X length(base) X width(height)

Cos(30) = [sqrt(3)] / 2 = 1.732050808... / 2 = 0.8660254068....

SecA = 1/CosA

Hence

SecASinA = SinA/CosA = 0

Then SinA = 0

A = 0 , 180, 360

It doesn't matter what the value of CosA is because if CosA = 0 , then the answer is undefined. 44e/.g/

SinA/CosA = SinA / 0 = Undefined.

195 degrees is in the third quadrant.

0 - 90 ; 1st quadrant

90 - 180 ; 2nd quadrant

180 - 270 ; 3rd quadrant

270 - 360 ; 4th quadrant.

Then the cycle repeats.

Firstly , it is 'pi' NOT 'pie'. 'pie' is what you eat ; apple pie.

Secondly ; make sure your calculator is in 'Radian' mode; NOT degree mode. Otherwise you will have an incorrect answer!!!!!

Type in ' Tan, Bracket 'pi', divide, '3', Close Bracket, equals (=) the answer should appear on your calculator screen as ' 0,01827908761,,,,'

NB Many students/people forget to correct the mode of their calculator, and thereby have an incorrect answer.

Degrees ; degrees mode

'pi' ; Radian mode.

To find the height of the building, we can use trigonometry. The angle of elevation (31°20') allows us to calculate the height above point A to the top of the building, while the angle of depression (12°50') helps us find the height from point A to the base of the building. By applying the tangent function for both angles and considering the height of point A (8.2 meters), we can derive the heights from these angles and find the total height of the building.

Tan(120) = Sin(120) / Cos(120) = [sqrt(3)/ 2 ] / [ -1/2]

[sqrt(3)/ 2 ] / [ -1/2]

Division of fractions.

[sqrt(3) / 2 ] X [ -2/1]

Cancel down by '2'

sqrt(3) / -1 =

• sqrt(3 = -1.732050808.....

The word ' trigonometry ' means ' measuring triangles '.

A three sided polygon is named a 'trigon', which we now name a 'triangle'.

'Metry' is to 'measure'.

Hence it follows that 'trigonometry' means ' measuring triangles'.

To find ( \cos 2A ) using the given ( \sin A = \frac{5}{13} ), we first use the Pythagorean identity to find ( \cos A ). Since ( \sin^2 A + \cos^2 A = 1 ), we have ( \cos^2 A = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} ). Thus, ( \cos A = \frac{12}{13} ). Using the double angle formula ( \cos 2A = 2\cos^2 A - 1 ), we get ( \cos 2A = 2\left(\frac{12}{13}\right)^2 - 1 = 2 \cdot \frac{144}{169} - 1 = \frac{288}{169} - \frac{169}{169} = \frac{119}{169} ).

Euler's coefficients in a Fourier series are derived from the process of projecting a periodic function onto the basis of sine and cosine functions. Specifically, for a function ( f(x) ) defined on an interval, the coefficients ( a_n ) and ( b_n ) can be calculated using integrals: ( a_n = \frac{1}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi nx}{T}\right) dx ) and ( b_n = \frac{1}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi nx}{T}\right) dx ), where ( T ) is the period of the function. The ( a_0 ) coefficient, representing the average value, is found using ( a_0 = \frac{1}{T} \int_{0}^{T} f(x) dx ). This systematic approach allows for the decomposition of any periodic function into its harmonic components.

The secant of an angle (2\theta), denoted as (\sec(2\theta)), is the reciprocal of the cosine of that angle. It can be expressed mathematically as (\sec(2\theta) = \frac{1}{\cos(2\theta)}). The value of (\sec(2\theta)) will depend on the specific angle (2\theta) and can be found using trigonometric identities or a calculator.