Sine, Cosine, Tangent and the Reciprocal Ratios

For the angle θ in a right-angled triangle as shown, we name the sides as:

• hypotenuse (the side opposite the right angle)
• adjacent (the side "next to" θ)
• opposite (the side furthest from the angle)

We define the three trigonometrical ratios sine θ, cosine θ, and tangent θ as follows (we normally write these in the shortened forms sin θ, cos θ, and tan θ):
  

To remember these, many people use SOH CAH TOA, that is:

Sin θ = Opposite/Hypotenuse,
Cos θ = Adjacent/Hypotenuse, and
Tan θ = Opposite/Adjacent

The Reciprocal Trigonometric Ratios

Often it is useful to use the reciprocal ratios, depending on the problem. (In plain English, the reciprocal of a fraction is found by turning the fraction upside down.)

Cosecant θ is the reciprocal of sine θ,
Secant θ is the reciprocal of cosine θ, and
Cotangent θ is the reciprocal of tangent θ

We usually write these as csc θ, sec θ and cot θ. (In some textbooks, "csc" is written as "cosec". Same thing.)
  
Important note: There is a big difference between csc θ and sin^-1x. The first one means "1/sin θ". The second one involves finding an angle whose sine is x. So on your calculator, don't use your sin^-1 button to find csc θ.
We will meet the idea of sin^-1x in the next section, Values of Trigonometric Functions.

The Trigonometric Functions on the x-y Plane

For an angle in standard position, we define the trigonometric ratios in terms of x, yand r:
  
Notice that we are still defining

sin θ as opp/hyp; cosθ as adj/hyp, and tan θ as opp/adj,

but we are using the specific x-, y- and r-values defined by the point (x, y) that the terminal side passes through. We can choose any point on that line, of course, to define our ratios.
To find r, we use Pythagoras' Theorem, since we have a right angled triangle:

Not surprisingly, the reciprocal ratios are defined similarly in terms of the x-, y- and r-values as follows:
  

Exponential Functions

Exponential functions have the form    f(x) = b^x

where b is the base and x is the exponent (or power).

If b is greater than 1, the function continuously increases in value as x increases. A special property of exponential functions is that the slope of the function also continuously increases as x increases.

Other calculators have a button labeled x^y which is equivalent to the ^ symbol.It is common to write exponential functions using the carat (^), which means "raised to the power". Computer programing uses the ^ sign, as do some calculators.

Example of an Exponential Function

Consider the function f(x) = 2^x.

In this case, we have an exponential function with base 2. Some typical values for this function would be:

x	-2	-1	0	1	2	3
f(x)	1/4	1/2	1	2	4	8

Here is the graph of y = 2^x.

Notice:

• That as x increases, y also increases.
• That as x increases, the slope of the graph also increases.
• That the curve passes through (0, 1). All exponential curves of the form f(x) = b^xpass through (0, 1), if b > 0.
• The curve does not pass through the x-axis. It just gets closer and closer to the x-axis as we take smaller and smaller x-values.

Logarithmic Functions

A logarithm is simply an exponent that is written in a special way.

For example, we know that the following exponential equation is true:

3² = 9

In this case, the base is 3 and the exponent is 2. We can write this equation inlogarithm form (with identical meaning) as follows:

log₃9 = 2

We say this as "the logarithm of 9 to the base 3 is 2". What we have effectively done is to move the exponent down on to the main line. This was done historically to make multiplications and divisions easier, but logarithms are still very handy in mathematics.

The logarithmic function is defined as:

f(x) = log_bx

The base of the logarithm is b.

The 2 most common bases that we use are base 10 and base e, which we meet inLogs to base 10 and Natural Logs (base e) in later sections.

The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and population prediction.

Example 1:

Write in logarithm form: 8 = 2³

Answer

log₂8 = 3
This just follows from the definition of a logarithm.

Example 2:

Write in exponential form: log₁₀1000 = 3

Answer

1000 = 10³
Once again, this just follows from the definition of a logarithm.

Example 3:

Find b if

Answer

The first line follows from the logarithm definition. The second line uses negative exponents.
Then we find the 4th root of both sides.

Exercises

1. Evaluate y = 9^x if x = 0.5

Answer

y = 9^0.5
y = 3

This is just a fractional index.

2. Express 8² = 64 in logarithmic form.

Answer

log₈64 = 2

3. Express log₁₁121 = 2 in exponential form.

Answer

11² = 121

4. Determine the unknown: log₁₀ 0.01 = x

Answer

Writing this in exponential form, we have: 10^x = 0.01. Solving this gives:

x = -2

5. Determine the unknown: log_b (1/4) = -1/2

Answer

Using the log laws, we can write this as:

So, using the rule for negative exponents gives:

Squaring both sides gives: b = 16.
Checking our answer, we have:
And as a logarithm, this can be written as:

log₁₆(1/4) = −1/2

Indices

Indices (or powers, or exponents) are very useful in mathematics. Indices are a convenient way of writing multiplications that have many repeated terms.

Example of an Index

5 is the base and

For the example 5³, we say that:

3 is the index (or power, or exponent).

5³ means "multiply 5 by itself 3 times".

[Or more accurately, "multiply 5 by itself repeatedly such that there are three 5's in the multiplication", or even better, "three 5's multiplied together". See a discussion on this at 

That is, 5³ means

5³ = 5 × 5 × 5 = 125

Examples of Integer Exponents

What happens if we have an index of 1, or maybe 0, or even -2?

Let's set up a pattern using our example above, so we can see what these special cases mean. As we continue this pattern, we are dividing by 5 to get each new line.

5⁴ = 5 × 5 × 5 × 5
5³ = 5 × 5 × 5
5² = 5 × 5
5¹ = 5
5⁰ = 1
5^-1 = 
5^-2 = 
5^-3 = 

Take note of the special cases

5¹ = 5,
5⁰ = 1, and
5^-1 = 

They are easy to mess up and they can make you lose sleep unnecessarily when you are doing algebra later.

In general , any number a, (except 0) raised to the power 1 is a.

a1 = a

Also, any number a, (except 0) raised to the power 0 is 1.

a0 = 1

And, any number a, (except 0) raised to the power -1 is 1/a.

Multiplying Numbers With the Same Base

We often need to multiply something like the following:

4³ × 4⁵

We note the numbers have the same base (which is 4) and we think of it as follows:

4³ × 4⁵ = (4 × 4 × 4) × (4 × 4 × 4 × 4 × 4)

We get 3 fours from the first bracket and 5 fours from the second bracket, so altogether we will have 3 + 5 = 8 fours multiplied together.

4³ × 4⁵ = 4³⁺⁵= 4⁸ (If anyone cares, the final answer is 65,536. :-)

In general, we can say for any number a and indices m and n:

Dividing Numbers with the Same Base

As an example, let's divide 3⁶ by 3²:

We cancelled out 2 of the threes on top and the 2 threes on the bottom of the fraction, leaving 4 threes on the top (and the number 1 on the bottom).

In general, for any number a (except 0) and indices m and n:

Raising an Index Expression to an Index

As an example, let's raise the number 4² to the power 3:

(4²)³ = 4² × 4² × 4²

From the multiplication example above, we can see that this is going to give us 4⁶. We could have done this as:

(4²)³ = 4^2×3 = 4⁶

In general, we have for any base a and indices m and n:

(am)n = amn

Raising a Product to a Power

Number example:

(5 × 2)³ = 5³ × 2³

In this case, with numbers, it would be better to perform the multiplication in brackets first and then raise our answer to the power 3. But when we are using letters in algebra, we cannot do such a thing and we need to know how to expand it out.

In general:

(ab)n = anbn

Raising a Quotient to a Power

Number example:

In general:

Summary of Index Laws

NOTE: There are no formulas for problems like a^m+ aⁿ = ...

This is because we can only add or subtract like terms (ones that have the same letter part). For example, this is okay:

5a² + 3a² = 8a²,

because we are adding like terms.

But we cannot do anything with the following expression:

5a³ + 3a⁷

Roots and Radicals

because these are unlike terms (not the same letter part).

We use the radical sign: 

It means "square root". The square root is actually a fractional index and is equivalent to raising a number to the power 1/2.

So, for example:

25^1/2 = √25 = 5

You can also have

Cube root:  (which is equivalent to raising to the power 1/3), and
Fourth root:  (power 1/4) and so on.

Key things to note:

If a ≥ 0 and b ≥ 0, we have:

However, this only works for multiplying. Please note that:

does not equal

(Try it with some real numbers on your calculator).

Also, this one is often found in mathematics:

This confuses a lot of students. But it just means:

1. Start with a number
2. Square it
3. Find the square root of the result
4. Finish with the number you started with

For example, start with 3.

Square it, you get 9.

Take the square root, you get 3, which is back where you started.

Why does it matter? Often we need to "undo" a square when solving an equation, so we find the square root of both sides. It's good to know what you are doing.