Forces at an AngleTrigonometry and Physics

The study of the many properties and relationships involving triangles.

Angles

Side lengths

Right triangles

Pythagorean Theorem

Sine, Cosine, and Tangent

Trigonometry

This was a law that showed us the relationships between the

sides of a right triangle.

Pythagorean Theorem

In Physics, this gave us the ability to combine perpendicular forces into a single value and find an object’s total acceleration.

These three values also focus on right triangles, but now give us

the relationship between each of the triangle’s sides based on

the other angles.

Sine, Cosine and Tangent

Before we define these new relationships, let’s make sure we understand how the different characteristics of the triangle are identified.

θ , H , A , O

Hypotenuse (H) – The longest side of the triangle and will always be located directly opposite the right (90°) angle.

Angle of interest (θ) – Slantedness of the hypotenuse when

compared to one of the other sides of the triangle

Sine, Cosine and Tangent

Opposite (O) – The length of the triangle side directly opposite the angle of interest

Adjacent (A) – The length of the triangle side touching the angle of interest (Other than the hypotenuse)

The relationships of Sine, Cosine and Tangent are fundamental

ratios (fractions) between two side of a right triangle based on a

given angle.

Sine (sin θ) – Ratio of the side opposite a given angle and the

hypotenuse

Cosine (cos θ) – Ratio of the side adjacent to a given angle and

the hypotenuse

Tangent (tan θ) – Ratio of the side opposite a given angle and

the side adjacent to that same angle.

Sine, Cosine and Tangent

In formula form, it looks a little something like this:

A popular pneumonic for remembering this is: SOH-CAH-TOA!

Sine, Cosine and Tangent

Now, if you know one of the non-right angles and any of the side

lengths, you can find the lengths of the other two sides.

Sine, Cosine and Tangent

Or if you know two side lengths, you could find the final side length and the angles of the triangle.

All you have to do is use the formula triangle to the left.

In physics, a lot of the forces we apply are at an angle and not

perfectly parallel to the surface we’re moving along and

therefor to the direction of our motion. Because of this, our

forces need to be split up into what we call components.

How does this apply to physics?

A component tells you how much of our total force is acting in the vertical (y) and horizontal (x) directions.

And by simply moving one of those components, we get a right triangle.

Let’s use the example of walking a

dog on a leash.

I pull up on Fido’s chain with some

force at an angle from the surface.

Now, let’s say that the force is 60 N

and the angle from the surface is 40°.

What are the vertical and horizontal components of force?

Example

White Board (conceptual)

Three sailboats are shown below. Each sailboat

experiences the same amount of force, yet has different

sail orientations. In which case (A, B or C) is the sailboat

most likely to tip over sideways? Explain.

White Board

A 400-N force is exerted at a 60-degree angle to move a

railroad car eastward along a railroad track. A top view of

the situation is depicted in the diagram. How much force is

pulling the cart to the right?

White Board

Consider the tow truck at the right. If the tensional force in

the cable is 1000 N and if the cable makes a 60-degree

angle with the horizontal, then what is the vertical

component of force that lifts the car off the ground?

White Board

A boy pulls his teddy around in a wagon. How hard is he

pulling the wagon forward, if he pulls with a force of 120N

on a handle that is angled at 40° from vertical?

White Board

A Man is pushing a 5 kg box with 400 N of force at an

angle of 60° from the frictionless surface. How much of his

force is moving the box forward?

What is the acceleration of the box?