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Logarithms, solving trig equations and functions ...

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Math 1060

Rules Logarithms Logarithms

Since perhaps it’s been a while, calculate a few logarithms just to warm up. 1. Calculate the following. (a) log3 (27) = (b) log9 (27) = (c) log3 ( 19 ) = (d) ln(e3 ) = (e) log(−100) = (f) ln(0) = Just as there are properties of exponents (like xa xb = xa+b ) there are properties of logarithms – in fact, this should be expected since exponentials and logarithms are so closely related. We’ll see how we can derive the properties of logarithms now. 2. Let’s pick a base just to simplify things – how about base 2. Let’s say x is some number, and let’s say X = log2 (x). Let’s say y is some other number, and Y = log 2 (y). Finally, let’s say Z = log2 (xy). (a) If Z = log2 (xy), write that as an exponential equation.

(b) If X = log2 (x), write that as an exponential equation.

(c) If Y = log2 (y), write that as an exponential equation.

(d) Take your answer to part (a), and substitute in your answers to parts (b) and (c). That is, 2Z = = (part (a) result) (parts (b) and (c) results) (e) Using rules of exponents, rewrite your result from part (d). 2Z = 2 (f) From part (e), what can you say about the relationship between Z, X, and Y ?

(g) Substitute back in the definitions of Z, X, and Y . What results is a rule of logarithms.

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Math 1060

Rules Logarithms

3. Summarize your result in a general form (since the base being 2 was irrelevant in the previous question).

We won’t prove the other rules explicitly here, but we’ll talk about why they make sense. 4. The rules of exponents state that ax+y = ax ay . (a) If f is the function f (x) = ax , that means that f (x + y) =

.

(b) In other words, with exponential functions, if you add inputs, that’s the same as outputs. (c) Since logarithms and exponential functions are inverses, that’s why it makes sense that with logarithmic functions, if you inputs, that’s the same as outputs. x

(d) The rules of exponents state that ax−y = aay . In other words, with exponential functions, if you subtract inputs, you outputs. (e) That means that with logarithms, if you

inputs, you

outputs.

(f) Try to write a rule of logarithms that was just described in the previous question.

5. In this question, we’ll develop our next rule. (a) What is loga (x · x)? You can use your result from Question 3 to rewrite this.

(b) What is loga (x · x · x)?

(c) What is loga (x4 )?

(d) What do you think loga (x38 ) should be?

(e) Write a rule to summarize.

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Math 1060

Rules Logarithms

6. Summarize the rules you’ve found so far, in Questions 3, 4.f, and 5.e. 1. 2. 3. Now we’ll practice using these a bit. For example, we could rewrite the expression log3 (x)−log 3 (y)+ 2 log3 (z) as follows:  2   x xz + log3 (z 2 ) = log 3 log3 (x) − log 3 (y) + 2 log3 (z) = log 3 y y 7. You try it. Use the rules of logarithms to write the following expressions as logarithms of one quantity with coefficient 1. (a)

1 ln x + ln 5 2

(b) log2 x + 4 log 2 (x + 1) −

(c) 5 ln x + 2 ln 3 − 3 ln

1 log2 (x − 1) 3

  1 y

8. What about the “other way?” Use the rules of logarithms to expand the following expressions so that there are no logarithms of products, quotients, or powers. (a) ln

p 3

x3 y

(b) log10

(c) ln



10 4x2

 √ x y (1 + x)3

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Rules Logarithms

9. Now use your critical thinking skills and the rules we’ve learned. Suppose ln x = 2, ln y = 3 and ln z = 6. Evaluate the following. (a) ln(xyz )

(b) ln(x2 y)

(c) ln



x3 √ z



Our last set of properties involves changing the base of a logarithm or exponential function. 10. Can you simplify 3x log 3 (5)? (First, try to change the expression in the exponent.)

11. What about ex ln(7)?

12. Let’s say I have 45 , and I want to write that as an expression with base 3 instead? That is, I want to write 45 = 3something . Let’s figure out how to do this. (a) Fill in the blank: 45 = 3log3 (

)

.

(b) Use rules of logarithms to rewrite your exponent.

13. In general, if you have ax , and you want to write that as bsomething , you can do this. Write down the rule below.

14. Now let’s see what the rule would be for logarithms, just by analogy. For exponentials, if you have ax , and you want to write this with base b, you the input by a factor of . 15. For logarithms, by analogy, if you have log a (x), an you want to write this using logarithms with base b, you should the output by a factor of . 16. The change of base formula for logarithms is:

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Math 1060

Rules Logarithms

Let’s use these a bit. 17. Write log 3 (5) as a logarithm with base 2.

18. Write ln(x) as a logarithm with base 10.

19. Simplify the expression log 3 (5) + log9 (5).

20. Write 5x as an exponential with base e.

21. Write 27 as an exponential with base 10.

22. Write xx as an exponential with base e.

23. Summarize the rules of logarithms so that you can remember them, making any notes to help you do so!

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Math 1060

Solving Trigonometric Equations Solving Trigonometric Equations

The easiest trig equations just involve a good knowledge of the unit circle. 1. Find a value for x such that sin(x) = −

√ 2 2

.

2. Find a value for θ such that cos(θ) = 12 .

√ 3. Find a value for t such that tan(t) = − 3.

In the above, you found a solution to those equations. When dealing with trig functions, however, there may be more than one solution. In fact, there’s usually an infinite number of solutions. Given an angle θ, we can write all angles that are coterminal with θ as “θ + 2πk, for any integer k.” For example, if we want to represent the set of angles {0, 2π, 4π, 6π, −2π, −4π, . . .}, we could just write “0 + 2πk, k ∈ Z” (that “k ∈ Z” stuff is mathematician shorthand for “k is any integer.”). 4. Find all values of x such that sin(x) = −

√ 2 . 2

5. Find all values of t such that tan(t) = 1.

6. Find all values of θ such that csc(θ) = 1.

If you have a more complicated trig equation, your main goal is to use algebraic techniques to transform it into something simple, like one of those above. √ 7. Solve for t: 2 cos t = −1.

Steven Pon

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Math 1060 8. Solve for t:

Solving Trigonometric Equations 3 + 2 sin t = sin t. 5

Sometimes we get tired of writing +2πk all the time. A common thing to do is to restrict our attention to solutions that lie in the interval [0, 2π ). 9. Find all solutions in the interval [0, 2π): 1 =

10. Find all solutions in the interval [0, 2π):

1 + 3 cos θ . 5 cos θ − 2

6 sec t + 2 = 2. 2 sec t − 1

Sometimes, some more complicated algebraic techniques might be required. Things like factoring, and then using the fact that AB = 0 =⇒ A = 0 or B = 0. Things like using the fact that x) 1 . Things like treating sin(x) as a single “thing” (which it is), and , or tan(x) = sin( sec(x) = cos(x) cos(x) factoring sin2 (x) − 2 sin(x) − 3 exactly the same way you would factor u2 − 2u − 3. √ 11. Find all solutions in [0, 2π): 2 sin2 t + 3 sin t = 0 (Try factoring the left hand side.)

Steven Pon

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Math 1060

Solving Trigonometric Equations

12. Find all solutions: 2 sin t cos t = sin t (Try moving all terms to one side and then factoring.)

13. Find all solutions in [0, 2π): 2 cos2 t + cos t − 1 = 0. (Try factoring it like a quadratic.)

14. Find all solutions in [0, 2π): sin t + tan t = 0. (Try rewriting tan(x), then factoring.)

15. Solve for θ: 2 sin2 θ − 3 sin θ + 1 = 0

16. Solve for x: tan x sec x +

Steven Pon

√ 2 tan x = 0

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Math 1060

Solving Trigonometric Equations

Sometimes your answers have to be expressed using inverse trig functions, since they won’t always work out nicely. 17. Find two solutions for x: 3 cos2 (x) + cos(x) − 2 = 0.

What if you had a more complicated expression inside a trig function? Something like tan(2x)? Hint: Let u = 2x, solve for u, and then substitute back to solve for x. 18. Find all solutions in [0, 2π): tan( x2 ) = 1.

19. Find all solutions: cos(2x) = −

Steven Pon

√ 2 2

.

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Math 1060

Solving Trigonometric Equations

You can also use trig identities to help out with simplifying equations. 20. Find all solutions to sin(2x) = cos x.

21. Solve sec2 x − 2 tan x = 4.

22. Find all solutions in [0, 2π) of 2 cot2 (x) + csc2 (x) − 2 = 0.

Steven Pon

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Math 1060

Trigonometric Functions Trigonometric Functions

Trigonometric functions are functions whose inputs are angles, and whose outputs are ratios. They’re especially useful for modeling periodic behavior. Let’s figure out exactly what they are. 1. Remember similar triangles? Let’s say you have the two right triangles below, and they have the same angles. What is x? x

5 θ

θ 2

6

2. Then what’s the relationship between

5 2

and 6x ?

Okay, so it seems that if you take a ratio of the two sides, the actual size of the triangle doesn’t matter, just the angles involved. Thus, we can create functions that have as input an angle in a right triangle, and that output a ratio of two sides. To describe these functions, we can name the sides: the opposite side is the one directly across from the angle, the hypotenuse is the longest side, and the adjacent side is the remaining side. Then we can define the following trig functions: sin(x) =

opp hyp

cos(x) =

adj hyp

tan(x) =

opp adj

csc(x) =

hyp opp

sec(x) =

hyp adj

tan(x) =

adj opp

The names are short for sine, cosine, tangent, cosecant, secant and cotangent. 3. Given the triangle below, and the angle θ in that triangle, what are the values of the trig functions if you input θ? (Hint: you might need the Pythagorean Theorem to help.) 5

θ 2

sin(θ) =

cos(θ ) =

tan(θ ) =

csc(θ) =

sec(θ ) =

tan(θ ) =

Of course, the largest angle that can appear in a right triangle is less than 90 degrees ( π2 radians). But we have many more angles than that. It would be nice to be able to input any angle into a trig function. We can generalize the idea of trig functions to encompass any angle. First, let’s look at angles of less than π2 radians, and see how we can view them in the unit circle.

Steven Pon

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Math 1060

Trigonometric Functions

4. Let θ = π3 . Draw θ in the unit circle. y

x

5. Now connect a vertical line from the point on the circle corresponding to θ down to the x-axis. Together with the radius of the circle and the x-axis, this line forms a triangle. What are the lengths of the sides of this triangle?

6. What is the sine of

7. Do the same for

π? 3

What is the cosine of

π ? 3

π . 6

8. Can you come up with a pattern that relates the sine and cosine to the coordinates corresponding to an angle? In general, the sine of an angle θ is the the cosine is the -coordinate.

-coordinate of the point corresponding to θ, and

That’s how we’re going to extend the definition of trig functions to any angle. Although only angles less than π2 can sit inside a right triangle, every angle on the unit circle has an x-coordinate and a y-coordinate. Thus, we can define the sine and cosine of any angle. )? 9. What is the sine of ( 2π 3

10. What is cos(− 3π )? 4 )? 11. What is sin( 7π 6

Steven Pon

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Math 1060

Trigonometric Functions

)? 12. What is cos( 7π 6 13. What is cos( π2 )?

)? 14. If you know the sine and cosine of an angle, you can say what its tangent is. What is tan( 7π 6

15. In fact, you can say what any of the trig functions are on that angle if you √ just know its sine and cosine. Let’s say θ is some angle such that sin(θ) = 13 , and cos(θ) = 2 32 . (Note that we’re not telling you what θ is...we don’t need to!) Then: tan(θ) =

csc(θ ) =

sec(θ ) =

tan(θ ) =

Now, conceivably, you can calculate trig functions for any angle, as long as you can figure out its coordinates on the unit circle. In general, that’s a tricky problem, which is partly why we memorize the coordinates for a few special angles and deal mostly with them, so we can focus on other parts of the theory. 16. What is the cosecant of

17. What is the tangent of

−11π ? 3

3π ? 2

18. Hopefully it’s also clear why trig functions are good for modeling periodic behavior. How do you know that all the trig functions are periodic?

Steven Pon

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