SOHCAHTOA

SOHCAHTOA is the analogy used to remember the three basic trigonometric factors, sine, cosine, and tangent.

What We've Learned

Trigonometry is used to find a missing side or angle of a right triangle.

Tangent

Tangent = Sine/Cosine

a circle with a radius of 1

the center is at (0,0)

hypotenuse always = 1

cos(θ) = x

sin(θ) = y

Graphs of Cosine

The cosine function is an up-down curve starting at 1 and declining until it gets to -1 at π radians, or 180°. The curve then goes back up until 1 at 2π radians, or 360°. The cosine graph is also is repeated every 360°, or 2π radians.

Graphs of Sine

The sine function creates a curve, going up from 0, that is repeated every 360°, or 2π radians. Its highest point is 1 at π/2 radians, or 90°, and its lowest point is -1 at 3π/2, or 270°

What is Trigonometry?

The Unit Circle

**Trigonometric Graphs**

**Trigonometry in Medicine**

**By Natalie Cassello**

sin(θ) = Opposite / Hypotenuse

cos(θ) = Adjacent / Hypotenuse

tan(θ) = Opposite / Adjacent

θ represents the angle that is being referred to.

Pythagoras Theorem

In a right angled triangle:

the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If a square is created on each side of a right triangle, then the area of the largest square will equal the sum of the areas of the two smaller squares.

This equation is used to find the third side of a right triangle, where c is the hypotenuse and a & b are the legs.

Triangulation

Orthopedics

**Medical Imaging**

Ultrasound Imaging

MRIs

Sinusoidal Graphs

So, using Pythagoras' Theorem

x² + y² = 1²

, but 1²=1.

(cos(θ))² + (sin(θ))² = 1

x² + y² = 1

Sine and Cosine

Degrees

Unit Circle in...

Radians

Example:

Example:

45°

1

(cos(45))² + (sin(45))² = 1

(√2/2)² + (√2/2)² = 1

(√2/2, √2/2)

Example:

1

(cos(π/4))² + (sin(π/4))² = 1

(√2/2)² + (√2/2)² = 1

(√2/2, √2/2)

π/4

tan(45°) = sin(45°)

/

cos(45°)

(√2/2)

/

(√2/2) = 1

We know that the radius is 1, so the hypotenuse is also 1. The angle given is 45°. To find the legs, we have to find cos(45°) and sin(45°). We use the equation for a unit circle, x² + y² = 1, or (cos(θ))² + (sin(θ))² = 1.

We found that the cosine measures √2/2, so the leg on the x axis is √2/2 and the sine measured √2/2, so the other leg is also √2/2.

We know that the radius is 1, so the hypotenuse is also 1. The radian given is π/4. To find the legs, we have to find cos(π/4) and sin(π/4). We use the equation for a unit circle, x² + y² = 1, or (cos(θ))² + (sin(θ))² = 1.

We found that the cosine measures √2/2, so the leg on the x axis is √2/2 and the sine measured √2/2, so the other leg is also √2/2.

45°

1

45°

In a 45°, 45°, 90° triangle, x = y.

x² + y² = 1

x² + x² = 1

2x² = 1

x² = 1/2

x = y = √(1/2)

√(1/2) = √(2/4) = (√2)/(√4) = √2/2

45°

45°

1

√(1/2)

√(1/2)

Trigonometric Graphs and the Unit Circle

Graphs of Tangent

135°

The Unit Circle Finger Trick

This method is used to memorize the coordinates of the most often used angles. To find the x- coordinate, count the number of fingers to the left and substitute that number for x in √x/2. To find the y-coordinate count the fingers to the right and substitute it for x.

90°

0°

45°

60°

30°

π/2

0

π/6

π/3

π/4

√x

2

Example: The coordinates of a 45° angle are (√2/2, √2/2) because there are two fingers on either side the the middle finger.

θ

The tangent function has a curve which is endless in both the positive and negative directions. As mentioned before, tan(θ) = sin(θ)/cos(θ). When sine, the numerator, is 0, tangent is also 0, so it intersects with 0. This occurs every π radians, or 180°. When cosine, the denominator, is 0, tangent is also 0, so it will be undefined. This graph creates a vertical asymptote, which is an infinite line avoiding a number, which is represented by the dotted line.

Triangulation is the process of finding a point from two existing points on a fixed baseline. When given two angles and a side length, a triangle can be created and the third vertex can be found.

D

θ¹

θ²

X

B

Z

To determine point Z, you need to find the distance of the line between Z and the point between X and Y on the fixed baseline. When given B, θ¹, and θ², use the formula:

Y

D =

B(sinθ¹)(sinθ²)

sin(θ¹ + θ²)

Example:

5

(0,0)

(3,0)

For this triangulation problem we are given the coordinates of two vertexes and the length of another side. We can easily find the length of the base by subtracting the x coordinates to find that the leg measures 3 units. Next, we can use the Pythagoras Theorem to determine the length of the last leg.

a² + b² = c²

3² + b² = 5²

9 + b² = 25

√b = √16

b = 4

Now that we know the height of the triangle, we can determine the coordinate of the the vertex by adding 4 to the y coordinate. In this case, the vertex is at (0,4).

Triangulation in Medicine

Triangulation is the process that comes after medical imaging. It is determining a specific location on the picture. For example, it can pinpoint a tumor growing in the brain.

This is an image of a knee created from an MRI. Triangulation was used to pinpoint the tear in the patient's meniscus.

Baseline

Angle

Angle

Trigonometry is used to determine the size of a plate needed to fix an open fracture or broken bone.

Example:

Length of bone = 25 cm

Plate needed = x

The length of the plate needed can be found using cosine.

Cosine = adjacent / hypotenuse

Cosine (30º) = x / 25

25 Cosine (30º) = x

3.856 = x

30º

The plate needs to be 3.856 cm long.

Sinusoidal graphs are graphs with consecutive curves. These include the sine and cosine graphs, which I mentioned earlier. These are used to monitor heart rate, breathing, and blood pressure.

Ultra sounds create an image on a computer from inside the body using sound waves. A transducer is slowly swept over the skin, which has gel on it. It then produces high-frequency sound waves that enter the body and transmits these waves to the computer once they bounce back off of the internal organs, fluids, and tissue. These sound waves are used to create an image. For pregnancies, they monitor the fetal sinusoidal heart rate to make sure the baby is breathing properly. The average pulse should be 120-160 bpm.

Sinusoidal sine waves are created from the high-frequency sound waves. The frequency is the amount of completed cycles per second. The amplitude is the maximum distance from 0, which is always 1 on a sine graph. The period is the time it takes to complete one cycle.

The average heart beat is known as the sinus rhythm and looks like this:

It's a graph of sine because it's a constant wave, or oscillation, repeated every 360º.

Medical imaging uses radiographic techniques to create an image of a part of the body. MRI stands for Magnetic Resonance Imaging, which is the process of a magnet lining up all of the hydrogen atoms in the body. Spinning protons then send the data to the computer where an image is developed. After a picture is created, doctors can look at it and use the process of triangluation to determine an unknown or unseen point.

CAT Scans

Computerized axial tomography is the process of creating detailed images of the inside of the body using X-rays and a computer. This technique is used mostly for diagnosis and monitoring. In the process, radiation passes through the body and is recorded on an electronic plate. Trigonometry is relevant to X-rays because the radiations must be angled correctly to get the most accurate image possible.

Angle of Injection

Heartbeat

Breathing

Blood pressure rises when the heart tightens and drops when the heart expands. This process of breathing is another example of sinusoidal graphs in medicine. The amplitude, in this case, would be the highest blood flow, and the period would be the heart rate.

Inhale

Exhale

In order to assure that the substance is being injected to the right place, the syringe must be angled appropriately. Most of the time it should be either 45° or 90°. The needle also can't be too deep because the liquid will be injected into the tissue instead of the muscle. Both the angle of the needle and its depth beyond the skin must be taken into account.

45°

2.15 cm

x cm

sin (45°) = x / 2.15

2.15 sin (45°) = x

1.829 = x

The needle should have a depth of 1.829 cm.

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