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Guide 3-1. Trigonometry: Math for 2-Dimensional Physics

This guide is intended to provide you with the trigonometry that you need to know in order to do physics in 2 dimensions.

Trigonometric Functions

Trigonometric or trig functions are simply ratios involving the sides of right triangles.

Consider the right triangle shown below. We're interested in relationships involving the angle, θ. Greek letters are frequently used to denote angles, and θ, pronounced theta, is the most common. People have found it convenient to define the ratios (see Table 1) and to give them the names sine, cosine, and tangent. Less commonly used functions are the reciprocal functions given in Table 2.

 Table 1: Commonly-used functions Name Function Ratio sine sinθ Lo/H cosine cosθ La/H tangent tanθ Lo/La
 Table 2. Reciprocal functions Name Function Ratio cosecant cscθ = 1/sinθ H/Lo secant secθ = 1/cosθ H/La cotangent cotθ = 1/tanθ La/Lo

Whenever you see one of these trig functions in your reading, remember the definitions. They're just ratios of two sides of a right triangle. They're handy, because the values depend only on the angle. All similar, right triangles with the same angle, θ, have the same values for the trig functions. Here's a mnemonic that people use to recall the ratios: SOHCAHTOA. Say it out loud and you won't forget it. It stands for "Sine = Opposite over Hypotenuse; Cosine = Adjacent over Hypotenuse; Tangent = Opposite over Adjacent.

 About inverses and trig functions Table 2 gives multiplicative inverses (reciprocals) of the trig functions. Another way to write a multiplicative inverse is the following: (tanθ)-1 = cotθ. Note that the entire function, tanθ in this case, goes in the parentheses. The notation, tan-1( ), means something entirely different than a multiplicative inverse. "tan-1(y/x)" means to find the angle whose tangent is (y/x). That is, θ = tan-1(y/x). Sometimes, this function is called the arctangent to avoid confusion with the term inverse. The tan-1 function on a calculator is the arctangent function. Note that there are several equivalent notations: θ = tan-1(y/x) θ = arctan(y/x) θ = atan(y/x) Similar considerations apply to the arcsine and arccosine functions.

In physics problems, you'll see angles like 30°, 45°, and 60° used frequently. That's because the values of trig functions for these angles are easy to retrieve mentally without resorting to using a calculator (or a set of tables in an earlier time). The common ones are given in Table 3.

 Table 3. Values of the trigonometric functions for special angles Angle Triangle sin cos tan Explanation 45° 1/√2 1/√2 1/1 Given that the legs are each 1 unit, the Pythagorean Theorem gives the hypotenuse. Note that sin45° = cos45° as a consequence of the fact that the legs of the triangle are equal. 60° √3/2 1/2 √3/1 An equilateral triangle with sides of 2 units is convenient to use. The altitude is the square root of 3 by the Pythagorean Theorem. The geometrical relationships are evident from viewing the triangle. Values of the trig functions for both 30° and 60° angles can be determined. 30° 1/2 √3/2 1/√3 0° 0 1 0 To get values for an angle of 0°, which of course is not a triangle at all, just imagine that the hypotenuse is swept down to give smaller and smaller values of θ. In the process, the hypotenuse approaches a length equal to that of the side adjacent to θ. At 0°, the hypotenuse coincides with the adjacent side, and the side opposite has shrunk to 0. 90° 1 0 undefined This is the opposite process as for 0°. Swing the hypotenuse up so that θ increases toward 90°. Note that the tangent approaches infinity. 37° 3/5 4/5 3/4 37° and 53° may seem like strange angles to use, but they happen to be--to 2 significant figures--the angles of a 3-4-5 triangle. 53° 4/5 3/5 4/3

Using the Trig Functions with Vectors

For now, the most important application of the trig functions is in determining the components of vectors. Given the direction angle and magnitude of a vector, the x- and y-components can be determined using the sine and cosine functions. Here's how the process works.

A. Vectors and Coordinate Systems

Suppose the vector A has a magnitude of A and an angle (also called direction angle) of θ. Note the following.

• A vector may be represented either with an arrow over the top, as in the diagram, or in boldface, like this: A. You must use one of the two notations in order to represent a vector without ambiguity.

• The magnitude, A, of the vector is represented in normal type without emphasis.

• In order to be able to state the direction of a vector, it must be placed in a coordinate system. The vector shown to the right is placed in an x-y coordinate system. Such Cartesian coordinates are standard. Directions on the surface of the Earth are also common.

• In an x-y coordinate system, the direction angle of a vector is measured counterclockwise from the +x axis. Direction angles therefore range from 0° to 360°.

B. Components of a Vector

The components of a vector are found by dropping perpendiculars from the tip of the vector to the x- and y-axes. Notice in the diagram that the components Ax and Ay are the legs of the right triangle adjacent and opposite to θ. Therefore,

cosθ = Ax/A and sinθ = Ay/A.

Rearranging, the components are given by the following.

Ax = Acosθ and Ay = Asinθ.

In case you're wondering why one would want to know the components, they're useful for adding vectors. This method is described in the Ch. 3 reading.

C. Signs of the Components

The components of a vector can be positive or negative depending on which quadrant the vector is in. Table 4 gives the signs; however, you can always figure them out just by looking at the vector in an x-y coordinate system and seeing whether the perpendicular dropped from the tip of the vector intersects the positive or negative extension of the axis.

You can also see that the signs of the trig functions depend on the quadrant into which the vector points. These are also included in the table. Again, it's not necessary to memorize these, as you can figure them out when you need them.

 Table 4 Sign of component Sign of trig function Quadrant x y sin cos tan I + + + + + II - + + - - III - - - - + IV + - - + -

D. Finding Magnitude and Direction Angle from Components

Let's suppose that the components of the vector are known and one wants to find the magnitude and direction angle of a vector. By the Pythagorean Theorem, the hypotenuse is square root of the sum of the squares of the two sides. Hence,

A = (Ax2 + Ay2)0.5.

The direction angle is obtained using the tangent function.

tanθ = Ay/Ax

The direction angle is then given by: θ = tan-1(Ay/Ax). In order to find the angle on a calculator, first calculate the ratio Ay/Ax. Then apply the tan-1 function.

Note that the inverse trig functions on a calculator don't tell you which quadrant the angle is in. This is something you determine from the context of the problem. Let's say, for example, that the vector A shown to the right has components of Ax = -2.5 and Ay = -4.5. Both components are negative, since the vector is in the third quadrant. The magnitude of the vector is

A = (Ax2 + Ay2)0.5 = ((-2.5)2 + (-4.5)2)0.5 = 5.1

The direction angle is found from

θ = tan-1((-4.5)/(-2.5)) = tan-1(1.8)

Now if you put 1.8 into your calculator and take the inverse tangent, the result is 61° to 2 significant figures.What the calculator has given you is the angle φ (pronounced fee) shown in the diagram. It's up to you to perform the following operation to determine the angle θ measured counterclockwise from the +x axis.

θ = 180° + φ = 241°

Let's look at another example. We rotate the vector into the fourth quadrant. Let's say the vector has the same magnitude as before, 5.1. It's not the same vector, though, because it has a different direction angle. So we'll give it a new name, B. Suppose the components are now Bx = 4.7 and By = -1.9. [You can verify that ((4.7)2 + (-1.9)2)0.5 = 5.1.]

The direction angle is found from

θ = tan-1((-1.9)/(4.7)) = tan-1(-0.40)

The calculator returns -22°, which is the angle φ as before. In order to determine the angle θ measured counterclockwise from the +x axis, you do the following.

θ = 360° + φ = 338°

Some Useful Identities

 There are some useful identities involving trig functions. Here are the most common ones. From the diagram to the right, you can see that sinθ = Ay/A. Now consider angle φ, which is the complement of θ. From an examination of the right triangle that includes φ, cosφ = Ay/A. That's because Ay is the leg adjacent to φ. Thus, we have sinθ = cosφ = cos(90 ° - θ) Here are three more identities that we give without proof: sin(-θ) = -sinθ cos(-θ) = cosθ sinθ / cosθ = tanθ Now consider the right triangle to the right. We have: sinθ = a/c and cosθ = b/c Square each trig function and add the squares to obtain: (sinθ)2 + (cosθ)2 = (a/c)2 + (b/c)2 = (a2 + b)2 / c2 The numerator (a2 + b)2 is equal to c2 by the Pythagorean Theorem. Therefore, we have: (sinθ)2 + (cosθ)2 = 1 Other useful identities involving the trig functions are found in Appendix 1 of your text.