Trigonometry

Cosine ( angle) = adjacent / hypotenuse.

The adjacent and the hypotenuse are the two lines that form the angle

The functions are periodic and so, given any value (within the range) the function can take the value several times, Graphing the function can help you determine secondary points at which the function takes a given value.

Sin(45 ) = 1/Sqrt(2) = 0.7071....

Cos(45) = 1/Sqrt(2) = 0.7071...

Tan(45) = 0.7071.../0.7071... Cancel down = '1'

Cosecant(45) = 1/Sin(45) = 1/ [1/sqrt(2) = sqrt(2) = 1.4142....

Secant (45) = 1/Cos(45) = 1/ [1/sqrt(2) = sqrt(2) = 1.4142....

Cotangent = Cos/Sin = 0.7071... / 0.7071... = 1.

The principal Trigonometric function is Sine(Sin).

It has a complementary function Cosine (Cos).

Bases on these two functions, are further trig. functions. they are.

Tangent(Tan) = Sin /Cos

Cosecant(CSC) = 1/Sin

Secant (SEC) = 1/Cos

Cotangent(COT) = Cos/Sin

The word 'Sine' comes from Latin , and means 'curve'. Initially you learn it between 0 - 90 degrees. It is a continuous 'wave', from infinity to infinity. .

The Sin/Cos functions only range from -1 to 1 through '0'.

The Tan function ranges from negative infinity to positive infinity, through '0'.

For a triangle inscribed in a circle.

At the circles centre, the radius may be '1' unit. If the radius is angles at 30 degrees from the horizontal(x) axis, then the vertical (opposite) lines perpendicular height of the triangle is exactly '1/2' ( 0.5), compared to the radius.

Hence the Sin( 30 degrees) is opposite / radius(hypotenuse) = (1/2)/1 = 1/2

All the othesr trig. functional answer are based on moving the radius from horizontal to vertical. with in the described circle. This is why it produces , such 'horrible' decimal figures.

Have a look in Castle's Four Figures Tables, rather than using a pocket calculator, and also graphs of the Sine/Cosine and Tangent curves.

Tan(90) is infinitely large. Hence it is deemed not solved.

Have a look at the graph of the Tan function.

Youwill see it passes through '0' at an angle of 45 degrees.

At 45 degrees , the graph line is steeper.

At 89 degrees it is almost a vertical line.

Hence a 90 degrees it is a vertical line and goes off the scale. Hence Tan(90) is unsresolved.

Tan(0) = 0

Tan (45) = 1

Tan(60) = 1.7320...

Tan(75) = 3.7320...

Tan(85) = 11.430...

Tan(89) = 57.289....

Tan(89.9) = 572.957...

Tan(89.999....) = 57295779.51

Notice how the Tan values becomes larger and larger with increasing angle.

Hence Tan(90) = unresolved; off the vertical scale.

Remember ' Csc(Cosecant)' = 1 /sin

Hence

Cos X 1/Sin = Cos/Sin = Cot (Cotangent)

Do NOT confuse with Tan = Sin /Cos. Notice the positions of trig. functions in the ratios.

Sec^(2) = 1 / Cos^(2)

Hence

Sec^(2) X Cos^(2) =

1/Cos^(2) X Cos^(2)

Cancel down by ' Cos^(2) '.

1/1 X 1 = 1

1 - Tan^(2) (The ANSWER) !!!!!

Factor

( 1 - Tan)(1 + Tan)

Tan(0) = 0

Remember

Tan(0) = Sin(0)/Cos(0) = 0/1

And Sin(0) = 0

Hence it follows that Tan(0) = 0

Trigonometry helps calculate planetary positions and celestial angles, providing mathematical support for astronomical calculations historically used in some astrological systems.

Tan )x degrees) = 1.66

x degrees) = Tan^(-1) 1.66

x = 58.9348.... degrees.

NB

Tan^(-1) on some calculators 9is shown as 'ArcTan'.

Let x = theta, since it's easier to type, and is essentially the same variable. Since tan^2(x)=tan(x), you know that tan(x) must either be 1 or zero for this statement to be true. So let tan(x)=0, and solve on your calculator by taking the inverse. Similarly for, tan(x)=1

NO!!!!

'Tan squared x = Tan^(2) [x]

Tan x squared = [Tan X}^(2)

Cosecant (CSC) = 1/Sine

Secant (SEC) = 1 /Cosine

Cotangent (COT) = 1 / Tangent = Cos/Sine

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