Introduction
In mathematics, trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. One set of trigonometric identities that are particularly useful in solving problems involving angles are the double and half angle identities. These identities allow us to express trigonometric functions of double or half angles in terms of trigonometric functions of the original angle. In this article, we will explore the concept of double and half angle identities and provide a worksheet to practice applying these identities.
Understanding Double Angle Identities
Double angle identities are formulas that relate trigonometric functions of twice an angle to trigonometric functions of the original angle. They can be derived from the sum and difference identities, which express the trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of the individual angles. The most commonly used double angle identities are:
Sine Double Angle Identity
The sine double angle identity states that:
sin(2θ) = 2sin(θ)cos(θ)
This formula tells us that the sine of twice an angle is equal to twice the product of the sine and cosine of the original angle.
Cosine Double Angle Identity
The cosine double angle identity states that:
cos(2θ) = cos²(θ) − sin²(θ)
This formula tells us that the cosine of twice an angle is equal to the difference between the squares of the cosine and sine of the original angle.
Tangent Double Angle Identity
The tangent double angle identity can be derived by dividing the sine double angle identity by the cosine double angle identity:
tan(2θ) = (2sin(θ)cos(θ))/(cos²(θ) − sin²(θ))
This formula expresses the tangent of twice an angle in terms of the sine and cosine of the original angle.
Using Double Angle Identities
Double angle identities are useful in a variety of trigonometry problems, such as finding the exact values of trigonometric functions or simplifying trigonometric expressions. To use these identities, follow these steps:
Step 1: Identify the Original Angle
Start by identifying the original angle for which you want to find the trigonometric function of the double angle.
Step 2: Select the Appropriate Double Angle Identity
Choose the double angle identity that corresponds to the trigonometric function you are interested in finding. For example, if you need to find the sine of the double angle, use the sine double angle identity.
Step 3: Substitute the Original Angle
Replace the θ in the double angle identity with the value of the original angle.
Step 4: Simplify the Expression
Use algebraic manipulations to simplify the expression and obtain the trigonometric function of the double angle.
Understanding Half Angle Identities
Half angle identities are formulas that relate trigonometric functions of half an angle to trigonometric functions of the original angle. They can be derived from the double angle identities by substituting θ with θ/2. The most commonly used half angle identities are:
Sine Half Angle Identity
The sine half angle identity states that:
sin(θ/2) = ±√[(1 − cos(θ))/2]
This formula tells us that the sine of half an angle can be expressed in terms of the cosine of the original angle.
Cosine Half Angle Identity
The cosine half angle identity states that:
cos(θ/2) = ±√[(1 + cos(θ))/2]
This formula tells us that the cosine of half an angle can also be expressed in terms of the cosine of the original angle.
Tangent Half Angle Identity
The tangent half angle identity can be derived by dividing the sine half angle identity by the cosine half angle identity:
tan(θ/2) = sin(θ)/(1 + cos(θ))
This formula expresses the tangent of half an angle in terms of the sine and cosine of the original angle.
Using Half Angle Identities
Half angle identities are useful in various trigonometry problems, including finding the exact values of trigonometric functions or simplifying trigonometric expressions. To use these identities, follow these steps:
Step 1: Identify the Original Angle
Start by identifying the original angle for which you want to find the trigonometric function of the half angle.
Step 2: Select the Appropriate Half Angle Identity
Choose the half angle identity that corresponds to the trigonometric function you are interested in finding. For example, if you need to find the sine of the half angle, use the sine half angle identity.
Step 3: Substitute the Original Angle
Replace the θ in the half angle identity with the value of the original angle.
Step 4: Simplify the Expression
Use algebraic manipulations to simplify the expression and obtain the trigonometric function of the half angle.
Double and Half Angle Identities Worksheet
Now that we have covered the theory and application of double and half angle identities, it's time to put your knowledge to the test with a worksheet. Below, you will find a set of problems that require the use of double and half angle identities. Try to solve them on your own before checking the provided solutions.
Problem 1
Find the exact value of sin(120°).
Solution:
Using the sine double angle identity, we have:
sin(2θ) = 2sin(θ)cos(θ)
Substituting θ = 60°, we get:
sin(120°) = 2sin(60°)cos(60°)
Since sin(60°) = √3/2 and cos(60°) = 1/2, we have:
sin(120°) = 2(√3/2)(1/2) = √3/2
Problem 2
Simplify the expression cos²(θ) − sin²(θ).
Solution:
Using the cosine double angle identity, we know that cos(2θ) = cos²(θ) − sin²(θ).
Therefore, the expression cos²(θ) − sin²(θ) simplifies to cos(2θ).
Problem 3
Find the exact value of cos(π/8).
Solution:
Using the cosine half angle identity, we have:
cos(θ/2) = ±√[(1 + cos(θ))/2]
Substituting θ = π/4, we get:
cos(π/8) = ±√[(1 + cos(π/4))/2]
Since cos(π/4) = √2/2, we have:
cos(π/8) = ±√[(1 + √2/2)/2]
Problem 4
Simplify the expression tan(θ/2) given that sin(θ) = 3/4 and cos(θ) = 1/4.
Solution:
Using the tangent half angle identity, we have:
tan(θ/2) = sin(θ)/(1 + cos(θ))
Substituting sin(θ) = 3/4 and cos(θ) = 1/4, we get:
tan(θ/2) = (3/4)/(1 + 1/4) = (3/4)/(5/4) = 3/5
Conclusion
Double and half angle identities are powerful tools in trigonometry that allow us to simplify expressions and find exact values of trigonometric functions. By understanding and applying these identities, you can solve a wide range of trigonometry problems with ease. Make sure to practice using the provided worksheet to reinforce your understanding and mastery of double and half angle identities.