Trigonometry

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.

The value of tan 0 is 0. This is because the tangent function is defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ)/cos(θ). At 0 degrees (or 0 radians), sin(0) is 0 and cos(0) is 1, resulting in tan(0) = 0/1 = 0.

Trig beacons, or triangulation beacons, are used for surveying and mapping purposes. They serve as reference points for geodetic and topographic surveys, helping surveyors determine precise locations and elevations. These beacons are often marked with distinct features to make them visible from a distance, facilitating accurate triangulation between multiple points. Additionally, they can assist in navigation and land development planning.

Sin^(2)[X] = 1 - Cos^(2)[X]

It is based on Pythagorean theorem .

Algebraically rearrange

Sin^(2)[x] + Cos^(2)[X[ = 1^(2) = 1

Note how it looks like the Pythagorean triangle h^(2) = a^(2) + b^(2) .

A sine bar is a precision measuring tool used in machining and metrology to measure angles. Its primary characteristics include two cylindrical rollers positioned at a fixed distance apart, typically 5 inches, which form the base for setting and measuring angles. The sine bar's accuracy depends on the flatness and parallelism of its surface, allowing it to be used in conjunction with gauge blocks to create precise angle setups. Additionally, it is made of high-quality materials to ensure durability and maintain precision over time.

2nd Quadrant.

The trigonometric value equal to cos 47° is sin(90° - 47°), which is sin 43°. This is based on the co-function identity in trigonometry, where the cosine of an angle is equal to the sine of its complement. Therefore, cos 47° = sin 43°.

It is the start of the second rotation, hence it is in the 1st quadrant.

Use the inverse/2nd function button on your calculator.

e.g.

Sin(angle) = 0.5

Then angle = Sin^(-1) 0.5 or ArcSin 0.5.

On your calculator make sure you are in DEGREE Mode.

Over the 'Sin' button there should be the symbol 'Sin^-1)'

So press ' Inver/2nd function/shift' as the case maybe for your calculator

Then 'Sin'

'0.5'

'='

Answer should be '30' Degrees.

Similarly for any angle and the two other trig functions (Cos) & (Tan).

Cosine(90) = 0

NB

Cosine(0) = 1

Cos(30) = 0.8669...

Cos(45) = 0.7071...

Cos(60) = 0.5

Cos(90) = 0

Cos(120) = -0.5

Cos(0135) = -0.7071...

Cos(150) = -0.8660...

Cos(180) = -1

NB

#1 ; refer to your (scientific) calculator

or

#2 ; refer to Castles Four Figures Tables.

NNB Note the negatives (-) between 90 & 180.

Sin or Sine is the ratio of the Opposite side Over The Hypoteneuse of any right angled triangle

inverse sin or inverse sine ( they mean the same thing ) uses the same ratio to find an unknown angle and can be written as sin-1

so if you know two sides ( opposite and Hypoteneuse ) then you can work out the sin...then you can either use a calculator to determine the inverse ( or angle ) or you could look up the sin in a booklet of sin values and determine the angle

Example : in the triangle ABC the line ab=4 ac = 5 and the angle abc =90

we could find the sin of the angle acb

sinacb= opp/hyp = 4/5 = 0.8

to find sin-1 of 0.8

calcultor press 0.8

press inv button or on newer calculators the 2nd functon button

then press the button marked sin

the display should then display 53.1301....or 53 to 2 sig figures

Cot (45 o) = Cos(45) / Sin(45)

Note the inversion of Cos and Sin.

Cos(45) = sqrt(2) / 2

Sin(45) = sqrt(2) /2

Applying division of fractions.

Cot(45) = Cos(45)/Sin(45) = [sqrt(2)/2] / [sqrt(2)/2] =>

sqrt(2)/2 X 2/Sqrt(2)

Note the change of sign and the inversion of terms.

Cancel down by '2'

sqrt(2)/1 X 1 / sqrt(2)

Cancel down again by 'sqrt(2)' .

1/1 x 1/1

= 1/1 = 1

Hence Cot (45 o) = 1

Yes, Microsoft Mathematics 4.0 can graph complex numbers and the unit circle. To graph complex numbers, you can enter them in the form (a + bi) (where (a) is the real part and (b) is the imaginary part) and plot them on the complex plane. To graph the unit circle, you can use the equation (x^2 + y^2 = 1), which represents all points with a distance of 1 from the origin. Simply input the equation in the graphing feature to visualize both the unit circle and any complex numbers on it.

To use Riemann sums for trigonometric functions, first define the interval over which you want to approximate the area under the curve, then divide this interval into ( n ) equal subintervals of width ( \Delta x ). Choose a sample point within each subinterval (either left endpoint, right endpoint, or midpoint) and evaluate the trigonometric function at these points. Multiply the function values by ( \Delta x ) and sum them up to estimate the total area. As ( n ) approaches infinity, this sum converges to the definite integral of the function over the specified interval.

Injuria sine damnum refers to a legal concept where a person suffers a legal injury or violation of rights without any accompanying actual damage or loss. An example would be if someone trespasses on your property, thereby violating your right to privacy, but does not cause any physical damage or loss to your property. In such cases, you could still pursue legal action for the infringement of your rights, even though there is no tangible harm done.

The inverse sine of -1, denoted as arcsin(-1), is the angle whose sine is -1. This value occurs at -π/2 radians (or -90 degrees) within the range of the arcsin function, which is typically from -π/2 to π/2. Therefore, arcsin(-1) = -π/2.