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This lesson will focus on introducing and practicing the trigonometric identities that relate the trigonometric values of an angle to the trigonometric values of the double-angle and half-angle.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Rules of the Quest Game and a Bonus

Zosia and her classmates entered a mathematical quest game. In this quest, there is an old leather map showing the path to various sacred mathematical sites. At each site, it is either solve the problem and move forward, or suffer the consequences of the unknown of the math universe.
A stream of water of a local fountain shoots into the air with velocity v at an angle theta with the horizont. It travels a horizontal distance of D=(v^2/g)2sin(2theta) and will reach a maximum height of H=(v^2/2g)sin^2(theta). Find H/D which helps determine the total width and height of the fountain.
External credits: @brgfx
At the very beginning of the quest, they were given a bonus task, which can be solved at the end of the quest. If solved successfully, they will get a huge amount of bonus points. What is the value of
Discussion

Presenting Double-Angle Identities

To be able to solve the tasks of the quest, the class first stopped at the infopoint to recall the Double-Angle Identities. These identities relate the trigonometric values of an angle to the trigonometric values of twice that angle.

Rule

Double-Angle Identities

The double-angle identities materialize when two angles with the same measure are substituted into the angle sum identities.

These identities simplify calculations when evaluating trigonometric functions of twice an angle measure.

Proof

Double-Angle Identities
Start by writing the Angle Sum Identity for sine and cosine.
Let and With this, becomes Then, these two formulas can be rewritten in terms of

Sine Identity

Approaching the first equation, the Commutative Property of Multiplication can be applied to the second term of its right-hand side. Then, by adding the terms on the right-hand side of this equation, the formula for is obtained.

Cosine Identities

Approaching the second equation, the Product of Powers Property can be used to rewrite its right-hand side. By doing this, the first identity for the cosine of the double of an angle is obtained.

Now, recall that, by the Pythagorean Identity, the sine square plus the cosine square of the same angle equals From this identity, two different equations can be set.
Next, substitute Equation (I) into the first identity for the cosine.
Substitute for and simplify
That way, the second identity for the cosine has been obtained. To obtain the third cosine identity, substitute Equation (II) into the first identity for the cosine.
Substitute for and simplify

Tangent Identity

To prove the tangent identity, start by rewriting in terms of sine and cosine.
Next, substitute the first sine identity in the numerator and the first cosine identity in the denominator.
Then, divide the numerator and denominator by
Finally, simplifying the right-hand side the tangent identity will be obtained.
Simplify right-hand side

Extra

Calculating

To calculate the exact value of these steps can be followed.

  1. To be able to use the double-angle identities, the angle needs to be rewritten as multiplied by another angle. Therefore, rewrite as
  2. Use the second formula for the cosine of twice an angle.
  3. Based on the trigonometric ratios of common angles, it is known that
Following these three steps, the value of can be found.

Simplify
Example

Searching for Clues

When the class got to the first mathematical sacred site, they were told to look for three clues. After eagerly searching the neighborhood, they found three parts to one task.
Three pieces of paper that say: 'Find sin(2theta)', 'cos(theta)=2/5','theta is between 0 and 90 degrees'
External credits: @brgfx
The task says to calculate knowing that and What exact value should the class get to be able to move on to the next site?

Hint

Find the value of by using one of the Pythagorean Identities.

Solution

In order to find the values of recall the Double-Angle Identity for sine.
The value of is known, but the value of is not. To find it, one of the Pythagorean Identities can be used.
Substitute with and solve for
Calculate root
By the definition of absolute value, can have two possible values.
It is given that is between and which is the first quadrant. Sine has positive values in the first quadrant, so the negative value can be disregarded.
Now that both sine and cosine of are known, the value of can be calculated.
Therefore, the value of is
Example

Finding Trigonometric Values to Determine the Password

At the second site, a steel safe awaits them. It is locked! Inside is the paper with the information needed to get to the third site. To open the safe, they must figure out the password.
A safe
External credits: @brgfx, @macrovector
The password is decoded sequentially by the integers that appear in the answers to the three given tasks. It is known that and
a Find the exact value of
b Find the exact value of
c Find the exact value of

Hint

a Use the Pythagorean Identity to find the value of
b Recall the Double-Angle Identity for cosine.
c Apply the Tangent Identity.

Solution

a Start by recalling the Double-Angle Identity for sine.
As can be seen, to find the value of the value of and should be known. It is given that Substitute that value into the Pythagorean Identity to calculate the corresponding value of
Calculate root
By the definition of absolute value, the cosine of can have two possible values — positive and negative.
It is known that which indicates that is in the second quadrant where cosine has negative values. This way the exact value of is found.
Now, the values of and can be used to calculate the value of
b In order to find the value of rewrite it by using the Double-Angle Identity for cosine.
Since the values of both and are known, substitute them into the equation and solve for
c Finally, to find the value of use the Tangent Identity.
Substitute the found values of and from the previous parts and then solve for
Discussion

Presenting Half-Angle Identities

Before moving to the next station, the class stopped at another infopoint to learn about the Half-Angle Identities.

Rule

Half-Angle Identities

The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied.

The sign of each formula is determined by the quadrant where the angle lies.

Signs of trigonometric ratios

These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.

Proof

Half-Angle Identities
First, write two of the Double-Angle Identities for cosine.

Sine Identity

Start by solving the first identity written above for
Solve for
Next, substitute for to obtain the half-angle identity for the sine.

Cosine Identity

Start by solving the second identity written at the beginning for
Solve for