Pythagorean Theorem and Quadratic Equation Introduction Discovered and developed by scientist and mathematician, Pythagoras (c. 570 BC – c. 495 BC),‘Pythagorean Theorem’ is a widely applied mathematical statement which illustrates the relation of the three sides of a right triangle that consists of two legs and a longest side, known as the hypotenuse. In equation, it is given by —

c2 = a2 + b2

where the variables ‘a’ and ‘b’ refer to the length measures of the right triangle’s legs while the variable ‘c’ pertains to the hypotenuse. Applications of Pythagorean Theorem are recognized in various fields of maths such as algebra, trigonometry, and calculus.

Problem 98: Buried Treasure

Ahmed has half of a treasure map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is ‘x’?

Solution:

By Pythagorean Theorem, x2 + (2x + 4)2 = (2x + 6)2

— x2 + 4×2 + 16x + 16 = 4×2 + 24x + 36

Combining similar terms,

— x2 + 16x – 24x + 16 – 36 = 0

Reduces to Quadratic Equation: x2 – 8x – 20 = 0

where the left side can be factored into (x – 10) * (x + 2) = 0

Then by Zero-Factor Rule: x – 10 = 0 and x + 2 = 0

Isolating ‘x’ in each equation gives x = 10 and x = -2

Taking the positive value, x = 10 paces

Since the Pythagorean Theorem is given by c2 = a2 + b2 then, each of the expressions ‘x’, ‘2x + 4’, and ‘2x + 6’ representing the sides of the right triangle which encloses the route to the buried treasure may be plugged into the Pythagorean equation such that (2x + 6)2 = x2 + (2x + 4)2 where ‘2x + 6’ paces refers to the measure of the longest side. Then expanding the binomials (2x + 6)2 and (2x + 4)2 yields 4×2 + 24x + 36 and 4×2 + 16x + 16, respectively. Upon combining like terms in the compound equation formed, the resulting equation turns out quadratic with x2 – 8x – 20 = 0.

By factoring the trinomial, x2 – 8x – 20 becomes the product (x – 10) * (x + 2). Through zero-factor property, each factor may be equated to zero to have x – 10 = 0 and x + 2 = 0, correspondingly. Solving completely, ‘x’ can be isolated on one side of each equation, becoming x = 10 and x = -2. It is logical to use positive values, so in this case, take x = 10. This means from Castle Rock to the place where the treasure is buried, Ahmed can walk 2*(10) + 6 or 26 paces to access the treasure or Vanessa can walk 10 paces heading north first then 2*(10) + 4 or 24 paces going east to be brought to the treasure spot.

Conclusion

Apparently, ‘Pythagorean Theorem’ proves useful in solving the specified problem which may be put into an illustration of a closed three-sided figure. Since there are distances covered northward and eastward, a 90-degree angle forms out of these segments, so that a right triangle is eventually created once the hypotenuse is drawn to connect the initial point (origin) and the final point (end). Such may be used in comparing the practicality of taking a straight route (along the hypotenuse) to the impracticality of taking two paths (shorter legs) which would consume more time of travel.

## Solve each of the trigonometric equations exactly # 21 25 29 33

In Exercises 19-36, solve each of the trigonometric equations exactly on 0 s 0 lt; 27.

TIAS

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= -V2

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NI

– 3 = 0

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28. sin?0 + 2sin0 + 1 = 0

29

2cos?0 = cos0

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The cosine

33.

in?0 + 2sin0 = 3

35. 4cos20 – 3 = 0

36. 4sin?0 = 3 + 4sin 0

Trigonometry

## Find two examples of parabolas in the real world or that model realworld phenomena Write a description for

Let f x x 2 4

The graph of f x x 2 4 as follows Let f x x 2 4 x 5 ax 2 bx c a The vertex h, k h b

4 2

2a

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Trigonometry

## Solve each of the trigonometric

In Exercises 1-18, solve each of the trigonometric equations exactly over the indicated intervals.

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Trigonometry

## What is tangent of the angle in standard position in which the terminal side passes through the point (3

Question

# What is:

tangent of the angle in standard position in which the terminal side passes through the point (3,

4)

cosine of the angle in standard position in which the terminal side passes through the point (-5, 12)

sine of the angle in standard position in which the terminal side passes through the point (-8, -15)???

Trigonometry

## Find the exact values of the remaining 5 circular functions (sine tangent cotangent secant and cosecant) for

Question

# Find the exact values of the remaining 5 circular functions (sine, tangent, cotangent, secant, and cosecant) for

angle θ, given that cosθ = 3/4 and θ is an angle in the 4 th quadrant. Find exact answers for this problem- no credit for approximations. Show work.

Trigonometry

## The height above the ground of a rider on a Ferris wheel can be modelled by the sine function h(x) = 25

Question

# The height above the ground of a rider on a Ferris wheel can be modelled by the sine function h(x) = 25

sin(x – 90°) + 27, where h(x) is the height, in metres, and x is the angle, in degrees, that the radius to the rider makes with the horizontal. Show on the table for one revolution of the Ferris wheel as follows:

Then predict and explain the values for a second revolution of the Ferris wheel.

Trigonometry

## Sketch the least positive angle and find the values of the six trigonometric functions3x+5y=0 x >

/=0

Question

# Sketch the least positive angle and find the values of the six trigonometric functions

3x+5y=0 , x /=0

8: Sketch the least positive angle and find the

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Trigonometry

## Using the Sine rule calculate the horizontal and

Question

Using the Sine rule calculate the horizontal and

vertical component forces. Then use any trigonometric calculation to find the resuant force ‘R’ and the angle at which it acts relative to the horizontal ‘r’

I find vectors impossible to understand, I don’t even know how to approach this problem. Is there any simple way of understanding this?

Trigonometry

## A mouse on a spring is moving with simple harmonic motion After a time tea the height age of the mouse above its

Question

# A mouse on a spring is moving with simple harmonic motion. After a time tea the height age of the mouse above its

rest position is given by the following formulaFind the height of the mass whenDo not round any intermediate compulsions round your answers to the nearest hundredth

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Trigonometry

## Need help with a trig exam see document attached

Thank you.

Question

# Need help with a trig exam, see document attached.

Thank you.

5.

1 x +0 y +2 z=−7→ x +2 z =−7

0 x+1 y +0 z=9 → y=9 z=t

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Trigonometry

## See question below and the photo for my comments

Perform the

Question

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Trigonometry

## Please show the step by step on how to solve it Thank you The Capital Wheel at the National Harbor is

Question

# Please show the step by step on how to solve it. Thank you.

- The Capital Wheel at the National Harbor is

a Ferris wheel with a diameter of 165 feet. If it were to spin at a rate of 2.1 revolutions per minute, what is the linear speed of a point on the outer edge of the wheel, in miles per hour?

Solve, finding ALL solutions. Express your answer exactly, using radians. Show work.

Trigonometry

## See attached questions and answers below from my grade 12 advanced functions text which includes trigonometry I

Question

# See attached questions and answers below from my grade 12 advanced functions text which includes trigonometry. I

understand the question and answer. For future problems, I know that it’s a good idea to break down given angles, i.e., cos (15) into the remarkable or known angles that are associated with a triangle and then apply the compound angle formulas.

Can someone please provide a list of the remarkable (or most often used in grade 12 Trig) angles please? Please give the angle values in degrees and radians. I searched google but too many hits came up!

If you can provide the name of the triangle the angles are from, that would be greatly appreciated too!

Trigonometry

## 1) Factor the expression then use fundamental identities to simplify

Question

**1) Factor the expression then use fundamental identities to simplify**

** **

**Cos^2xsec^2x-cos^2x**

**2) Perform the subtraction then use the fundamental identities to simplify **

a) 1/secx+1 – 1/secx-1

b) tanx- sec^2x/tanx

Really need help understanding how to do these problems?

Trigonometry