Trig Basics

7/30/2019 Trig Basics

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ACTTest: Trigonometry, Logarithms, and Matrices

(Adapted from OnLineMathLearning.com and fromSparkNotes, in accord with Terms and Conditions)

Lets start with trigonometry. You only have to know a few basics to do well on theACT trig questions. The topics of trig covered by the ACT are:

1. Sine, cosine, tangent 2. Solving Triangles3. Trigonometric Identities 4. Trigonometric Graphs

Definitions

Cosecant Hypotenuse over OppositeSecant Hypotenuse over Adjacent

To understand trig functions better, construct a right triangle that includes theangle in question. Opposite means the side facing the angle; adjacent means theside thats next to the angle, but isnt the hypotenuse (the side opposite the 90angle). Say you have the following right triangle:

There are two right triangles you should know inside-out with before sitting down totake the ACT: the 45-45-90 degree triangle and the 30-60-90 degree triangle. Youshould, for example, memorize the values of the sine, cosine, and tangent for theangles in these triangles, and also for the angles 0 and 180. ACT trig questionsoften involve these angles, and if you know the values in the table below, you canspeedily find the answer.

Solving TrianglesOnce you understand the trigonometricfunctions of sine, cosine, and tangent, youshould be able to use these functions to

solve a triangle: if you are given some

information about a triangle, you can use thetrigonometric functions to figure out thevalues of other angles or sides of the triangle.

Sine Opposite over HypotenuseCosine Adjacent over HypotenuseTangent Opposite over Adjacent

Angle Sine Cosine Tangent

0 0 1 0

30 1/2

45 1

60 1/2

90 1 0 Undefined


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In this problem, if you are familiar with the

30-60-90 degree triangle, youll alreadyknow the ratio of the sides in such atriangle to one another. But you can also

use trig to solve for . You are given the

measure of , as well as the length of. The diagram also shows that this

triangle is a right triangle. You can use this

information to solve for if you can figure

out which trigonometric function to use. You have to find the value of side ,which stands opposite the angle you know. You also know the value of the

hypotenuse. To figure out , then, you can use the trig function that uses bothopposite and hypotenuse, which is sine. From the chart of the values of criticalpoints, you know that sin 30 = 1/2. To solve:

Another favorite ACT problem is to combine the Pythagorean theorem with trigfunctions, like this:

What is the sine of in right triangleABCbelow?

To find the sine of , you need to know the value of the side opposite and thevalue of the hypotenuse. The figure gives the value of the hypotenuse, but not of

the opposite side. However, since the figure does provide the value of , you can

calculate the value of the opposite side, , by using the Pythagorean theorem.

Now that you know the value of , you can solve for sineA:

For example, what is the length

of in the triangle below?


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Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that holdstrue for all angles. For the ACT test, trigonometric identities, on those few occasionswhen they come up, will be helpful in situations when you need to simplify atrigonometric expression. The two identities you should know are:

1. 2.

Radian mathematics:A radian is a central angle, for whichlengths of its arc and radius are equal ( AmB = AO in figure). So,a radian measure of any angle is a ratio of a length of a circle arcto a circle radius.

There are 2 radians in acircle, since the ratio of thecircumference to the radiusis 2.

Trigonometric GraphsThe ACT will include one or two questions covering the graphs of the trigonometric

functions. You should be able to match each graph with each function, and youshould know when the different functions reach their highest point and lowest point.

Graph ofy= sinx

Graph ofy= cosx


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Graph ofy= tanx

Stretching the Trigonometric GraphsIn addition to knowing the graphs of the trigonometric functions, you should alsoknow how the graphs can be stretched vertically or horizontally. Vertical stretchesaffect the graphs amplitude, while horizontal stretches change the period.

Stretching the AmplitudeIf a coefficient is placed in front of the function, the graph will stretch vertically: itshighest points will be higher and its lowest points will be lower. Whereas thefunction y= sin x never goes higher than 1 or lower than 1, the function y= 3sin xhas a high point of 3 and a low point of3. Changing the amplitude of a functiondoes not change the value ofxat which the high and low points occur. In the figurebelow, for example, y = sin x and y = 3 sin x both have their high points when x

equals 3/2 and /2.

The amplitude of a trigonometric function is equal

to the absolute value of the coefficient thatappears before the function. The amplitude ofy = 2 cos x is 2, the amplitude ofy= 1/2 sinxis1/2, and the amplitude ofy= 2 sinxis 2.

Stretching the PeriodIf a coefficient is placed before thexin a trigonometric function, the function isstretched horizontally: its curves become steeper or less steep depending on the

coefficient. The curves ofy= sin 3xare steeper than the curves ofy= sin (x/2).This coefficient doesnt affect the amplitude of the function in any way, but it does

affect where on thex-axis the function has itshigh and low points. The figure below showshow changing the period affects a sine curve.

The ACT may test your knowledge of periodsby presenting you with a trig function thathas a period coefficient and asking you for the


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smallest positive value where the function reaches its maximum value. Threeexamples:

1. What is the smallest positive value forxwhere y= sin 2xreaches its maximumvalue?

2. What is the smallest positive value forxwhere y= cos 2xreaches its maximum value

3. What is the smallest positive value forxWhere y = cos3x reaches its max. value?

The answer to the first question is 45o or /4, the answer to the 2nd is 180o or ,the answer to the 3rd is 120o, or 2/3. (Note that zero is another value where cos2x reaches its maximum, but zero is not a positive value and is therefore not theanswer.)

You should now be able to answer this question:

If y = , then which of the following functions has 3 times the amplitudeand one-third of the period?

A. y =

B. y =

C. y =

D. y =

E. y =Answer: C

Angles more than 90 degrees. Note that angles are measured counter-clockwisefrom the positive x axis. Hence we can represent an angle = 240O in the thirdquadrant (i.e. an angle in between 180 and 270 degrees) as shown in the diagrambelow:

In this diagram, Ive represented as located within the triangle drawn in

the 3rd

quadrant, but actually it stretchescounter-clockwise from the positive x axisto the hypotenuse of that triangle.

By visual inspection, you can see whathappens to trigonometric functions thatare not in the first quadrant of a circle:

Angle in first quadrant: sin and cosineare positive; tangent is positive


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Angle in 2nd quadrant: sin is positive, cosine negative; tangent is negative

Angle in 3rd quadrant: sine is negative, cosine is negative, tangent is positive

Angle in 4th quadrant: sine is negative, cosine is positive, tangent is negative

Trig. Exercises1. (sinA)2 = ?A. 1 - (cosA)2 B. (cosA)2 - 1 C. (tanA)2 D. 1 - (tanA)2 E. (tanA)2 - 1

2. If cosA = y/z and sinA = x/z, then tanA = ?F. z/x G. z/y H. xy/z J. x/y K. y/x

3. Ifxrepresents a real number, what is the greatest possible value of 4 sin 2x?A. 2 B. 3 C. 4 D. 6 E. 12

4. Angles A and B each have measurements between 0 and 45.If cos A = sin B, what is the sum of Angle A and Angle B?F. 15 G. 30 H. 45 J. 90 K. 180

5. If cos = -3/5 and < < 3/2, then tan =F. 3/5 G. -3/4 H. -3/5 J. 5/4 K. 4/3

(To solve this last problem, you may refer to the 4-quadrant diagram above. Note that

the x coordinate is -3 and the y coordinate is -4.)

Exercise Answers: 1A 2J 3C 4J 5K

LogarithmsLogarithms (like matrices) rarely appear in ACT Math. But sometimes they do, andyou should know how to handle them. Logarithmic functions are inverses ofexponential functions. That is:

The exponential equation x = ab is equivalent to the logarithmic equation logax = b.

This relationship between logs and exponents is all you need to know in order toanswer a logarithm question on the ACT. If you see

logx16 = 4, then you know that x4 = 16, and therefore x = 2.

You will be able to use this second, exponential form to answer the question.

Other key rules about logarithms: log(xy) = log x + log y log (x/y) = log x log y

and logxcn = n . logxc


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You can now see how to break down a logarithmic expression into bite-sizedcomponents:

logabc

d3= log (abc) logd3 = log a + log b + log c logd3

= log a + log b + 2 log c 3 logd

OK, now exercise what youve learned by solving for x in a couple of equations:

x = log84 + log816 x = ?

Answer:x = log864 = 2

log224 - log23 = log5x x=?

Answer:log224 - log23 = log5x (begin by using the division rule for logarithms)log28 = log5x

3 = log5xx = 125

Lets try one more. Please solve for x:

= 4 Answer: = 4

= 4

) = 4

= 4

= 1

= 3

x = 8

Matrices.Matrix problems occur rarely on an ACT test, but you should be prepared toencounter them.


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You can add or subtract matrices. For example,

What isA + B? To answer this question, you simply add the corresponding entries

inA and B. The entries in the first row are 2 + (4) = 2 and 0 + 1 = 1. Theentries in the second row are 3 + 6 = 9 and (5) + 3 = 2. So the resultingmatrix is:

If the question had asked you whatAB is, then you would simply subtract theentries in B from the corresponding entries inA.

Here is another sum of two matrices, but the second one is multiplied by the

number 2:

0 1

1/3 -1

+ 2

1 -1

2/3 -2

=

2 -1

5/3 -5

Matrix multiplication:

Find C = A B

Step 1 : Multiply the elements in the first row of A with the corresponding elementsin the first column of B. Add the products to get the element C11 The subscript

11 refers to the element in the first (i.e. leftmost) position of the first (i.e. thetop) row.

Step 2 :Multiply the elements in the first row of A with the corresponding elements in the

second column of B. Add the products to get the element C12


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Step 3 :Multiply the elements in the second row of A with the corresponding

elements in the first column of B. Add the products to get the element C21

Step 4 : Multiply the elements in the second row of A with the correspondingelements in the second column of B. Add the products to get the element C22

After studying the above, see if you can multiple these two matrices:

0 1

1/3 -1

1 -1

2/3 -2=

Answer:

2/3 -2

-1/3 5/3


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SAT II Test Preparation

Trigonometric Formulas

A trigonometric identity is an equation involving trigonometric functions that holds true

for all angles. These identities are commonly called Pythagorean identities, becausethey come from the Pythagorean theorem.

In general, the Math IIC will test your knowledge of the trigonometric identities bygiving you a complex expression that you have to simplify. These questions have moreto do with memorizing the identities and being good with algebraic substitution than

they do with the theoretical concepts of trigonometry.

For example:

What is ?

To solve a problem like this, use the trigonometric identities to simplify the

trigonometric expression into sines and cosines. After you have simplified theexpression using the identities, it is quite likely that the expressions will simplify furtherthrough the canceling of terms. The simplification of the expression in the example

questions proceeds as follows.

Simplify the mess given to you by the problem, and you get sin2

.Heres another example:

What is tan2 cos2 + 1 sin2 ?


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Other Trigonometric IdentitiesAbout once a test, you may encounter a particularly complicated trigonometricexpression. Such expressions can usually be solved using the trigonometric identitiesweve already covered, but the process will be laborious and difficult. However, if you

know the special trigonometric identities below, these questions will be much easier andless time-consuming.We list the special trigonometric identities in the order of usefulness, starting with the

ones most likely to be used on the test.

Sum and Difference FormulasThere are three identities for the sum of two angles:

There are also three identities for the difference of two angles:

Try to use these identities in a sample problem:

Simplify the following expression: sin 2x+ cos2x sec2x+ tan2x.

Using the identities sin2x+ cos

2x= 1 and 1 + tan

2x= sec

2x, the expression simplifies

in the following way:

Heres another example to practice with:

If sin a = cos b = 3/5 and a and b are both in the second quadrant, what is cos (ab)?

If sin a = 3/5 and the angle is in the second quadrant, then cos a = 4/5. Also, if cos b =

3 /5 and b is in the second quadrant, then sin b =4/5. See the figure below:


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Now you can use the subtraction identity to see that

= (4 /5)(3/5) + (

3/5)(4/5) =

12/25 +12/25 =

24/25.

If you are familiar with the formula for cos(ab) and remember the Pythagorean triple3-4-5, you can save time on a question like this. Otherwise, you have to calculate the

arcsine of3/5 and the arccosine of3/ 5 in order to find and , respectively. Then

you have to subtract from and take the cosine of the resulting angle.

Double Angle Formulas

There are three double angle formulas.

Side Angle Side

This is the formula for finding a third side of a triangle when you know the lengths ofthe two other sides and their included angle:

side c = a + b -2ab cos(C)

Half Angle Formulas

Of all trigonometric identities, the six half-angle formulas appear most infrequently onthe Math IIC.

Trig Basics

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