Fundamental Trigonometric Identities - Pre-Calculus

Write the half angle identity for sine.

Since we are given , the angle is equal to . Set these two angles equal to each other and solve for .

Substitute this value into the formula.

We have:

and :

(1)-(2) gives:

Knowing from the above formula that:( take a=b in the formula above)

This gives:

We note first, using trigonometric identities that:

This gives:

Since,

We have :

We can write the above sum as :

From the given fact, we have :

and we have : .

This gives :

Using trigonometric identities we have :

and we know that:

This gives us :

Hence:

We will use the given formula :

We have in this case:

Since we know that :

This gives :

We can write the above sum as :

From the given fact, we have :

This gives us :

Therefore we have:

Of the six trigonometric functions, four are odd, meaning . These four are:

• sin x
• tan x
• cot x
• csc x

That leaves two functions which are even, which means that . These are:

• cos x
• sec x

Of the aforementioned, only is incorrect, since secant is an even function, which implies that

By the double angle formula

By the double angle formula

By the double-angula formula for cosine

For this problem

By the double-angle formula for the sine function

we have

thus the double angle formula becomes,

When trying to identify equivalent equations that use trigonometric functions it is important to recall the general formula and understand how the terms affect the translations.

The general formula for sine is as follows.

where is the amplitude, is used to find the period of the function , represents the phase shift , and is the vertical shift.

This is also true for,

.

Looking at the possible answer choices lets first focus on the ones containing sine.

has a vertical shift of therefore it is not an equivalent function as it is moving the original function up.

has a phase shift of therefore it is not an equivalent function as it is moving the original function to the right.

Now lets shift our focus to the answer choices that contain cosine.

has a vertical shift down of units. This will create a graph that has a range that is below the -axis. It is important to remember that has a range of . Therefore this cosine function is not an equivalent equation.

has a phase shift to the right units. Plugging in some values we see that,

,

.

Now, looking back at our original function and plugging in those same values of and we get,

,

.

Since the function values are the same for each of the input values, we can conclude that is equivalent to .

The expression is a double angle identity that can also be rewritten as:

Replace the value of theta for .

The correct answer is:

A useful trigonometric identity to remember is

If we plug in into this equation, we get

We can divide the equation by 2 to get

A useful trigonometric identity to remember for this problem is

or equivalently,

If we substitute for , we get

We have using the given result:

This gives us:

Hence :

Using trigonometric identities we have :

We know that :

and

This gives :

Using trigonometric identities we know that:

Letting and in the above expression we have:

We also know that:

and .

This gives: