Sum and Difference Identities of Sine and Cosine

The Organic Chemistry Tutor

30 Jan 202121:37

EducationalLearning

32 Likes 10 Comments

TLDRThis video tutorial explores the essential trigonometric sum and difference formulas for sine, cosine, and tangent functions. The instructor explains how to use these formulas to solve problems involving specific angles like 15 degrees, 105 degrees, and more, using real trigonometric values such as √2/2, √3/2, and others. The session includes step-by-step methods to break down complex angles into known values on the unit circle, simplifying them through algebraic manipulation to find exact values. Additionally, examples are provided to demonstrate practical applications of these formulas in evaluating trigonometric expressions and solving equations.

Takeaways

📐 **Sum and Difference Formulas**: The video discusses the trigonometric sum and difference formulas for cosine and sine, which are essential for solving various trigonometric expressions.
🔍 **Cosine Formulas**: The formula for cosine of the sum or difference of two angles is given as `cos(α ± β) = cos(α)cos(β) ± sin(α)sin(β)`, with the sign depending on whether the sum or difference is taken.
🔍 **Sine Formulas**: The formula for sine of the sum or difference of two angles is `sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)`, with the sign changing based on the operation (sum or difference).
📈 **Tangent Formulas**: The tangent formula for the sum or difference of two angles is `tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β))`, with the sign in the denominator matching the operation sign.
🧮 **Evaluating Specific Angles**: The video provides methods to evaluate trigonometric functions for specific angles, such as 15°, 105°, and others, by breaking them down into known angles on the unit circle.
🔢 **Values of Trigonometric Functions**: Common values for trigonometric functions at specific angles (e.g., sin(45°), cos(30°)) are used to simplify expressions.
🤔 **Understanding Sign Changes**: It's crucial to understand how to change the signs in the formulas based on whether you are adding or subtracting angles.
🛠️ **Using Formulas for Calculation**: The video demonstrates the application of the sum and difference formulas to calculate the sine and cosine of angles that are not directly available on a unit circle.
🤓 **Mastering the Formulas**: Emphasizes the importance of knowing the sum and difference formulas for solving complex trigonometric problems.
📉 **Example Calculations**: Several examples are worked through in the video, illustrating step-by-step calculations using the sum and difference formulas.
📚 **Practical Application**: The video script serves as a practical guide for students or individuals looking to understand and apply trigonometric sum and difference formulas.

Q & A

What are the sum and difference formulas for cosine?
-The sum formula for cosine is cos(α + β) = cosα * cosβ - sinα * sinβ, and the difference formula is cos(α - β) = cosα * cosβ + sinα * sinβ.
How does the sign in the sum and difference formulas for sine work?
-The sign in the sine sum and difference formulas remains the same as the sign on top. For example, sin(α + β) = sinα * cosβ + cosα * sinβ, and sin(α - β) = sinα * cosβ - cosα * sinβ.
What is the formula for tangent in terms of the sum and difference of two angles?
-The formula for tangent of the sum of two angles is tan(α + β) = (tanα + tanβ) / (1 - tanα * tanβ), and for the difference is tan(α - β) = (tanα - tanβ) / (1 + tanα * tanβ).
How can you find the value of sine for 15 degrees using sum and difference formulas?
-You can find the value of sine for 15 degrees by using the formula for sine of the difference of two angles: sin(45 - 30) = sin45 * cos30 - cos45 * sin30, which simplifies to (√2/2) * (√3/2) - (√2/2) * (1/2) = √6/4 - √2/4.
What are the values of sine and cosine for 30 and 45 degrees?
-The value of sine for 30 degrees is 1/2, and for 45 degrees is √2/2. The value of cosine for 30 degrees is √3/2, and for 45 degrees is also √2/2.
How do you find the value of sine for 105 degrees?
-You can find the value of sine for 105 degrees by using the formula for sine of the sum of two angles: sin(60 + 45) = sin60 * cos45 + cos60 * sin45, which simplifies to (√3/2) * (√2/2) + (1/2) * (√2/2) = √6/4 + √2/4.
What is the cosine of 2π/3 minus π/6 radians?
-The cosine of (2π/3 - π/6) radians can be found using the difference formula for cosine: cos(2π/3) * cos(π/6) + sin(2π/3) * sin(π/6), which simplifies to (-1/2) * (√3/2) + (1/2) * (1/2) = -√3/4 + 1/4.
What is the value of tangent for 15 degrees?
-The value of tangent for 15 degrees can be found using the formula for tangent of the difference of two angles: tan(60 - 45) = (tan60 - tan45) / (1 + tan60 * tan45), which simplifies to (√3 - 1) / (1 + √3), and after rationalizing the denominator, it equals 2 - √3.
How do you use the sum and difference formulas to evaluate trigonometric expressions?
-You can use the sum and difference formulas to evaluate trigonometric expressions by first identifying the correct formula based on the operation (sum or difference) and the trigonometric function (sine or cosine) involved. Then, you find the values of the trigonometric functions for the angles given and apply the formula to find the result.
What is the cosine of 5π/12 radians?
-The cosine of 5π/12 radians is the same as the cosine of 75 degrees. Using the sum formula for cosine with angles 45 and 30 degrees, cos(45) * cos(30) - sin(45) * sin(30), which simplifies to (√2/2) * (√3/2) - (√2/2) * (1/2) = √6/4 - √2/4.
How do you find the exact value of the expression cos(70) * cos(40) + sin(70) * sin(40)?
-You can find the exact value by recognizing it as the cosine of the difference of two angles: cos(70) * cos(40) + sin(70) * sin(40) is equal to cos(70 - 40), which is cos(30), and the value of cos(30) is √3/2.
What is the sine of 4π/12 or 60 degrees?
-The sine of 60 degrees, which is the same as 4π/12 radians, is √3/2.

Outlines

00:00

📚 Sum and Difference Formulas for Trigonometry

This paragraph introduces the sum and difference formulas for cosine, sine, and tangent. It explains how to use these formulas to find the values of trigonometric functions for specific angles. The cosine formula is detailed with examples of both addition and subtraction, emphasizing the importance of the sign in the formula. The sine formula is then presented, noting that the sine function does not change with the sign. The tangent formula is also explained, with an example of how to adjust the signs based on the given angle. Finally, the paragraph demonstrates how to use these formulas to calculate the sine of 15 degrees using the angles 45 and 30 degrees.

05:00

🔢 Evaluating Trigonometric Expressions with Sum and Difference Formulas

The second paragraph continues the discussion on sum and difference formulas, focusing on how to apply them to evaluate specific trigonometric expressions. It provides step-by-step examples for finding the sine of 105 degrees, cosine of two pi divided by three minus pi over six, and cosine of 5 pi over 12. Each example shows the process of identifying the appropriate formula, performing the necessary calculations using known values of sine and cosine for specific angles, and simplifying the expression to find the final answer.

10:01

🤔 Using Sum and Difference Formulas for Cosine and Sine

This paragraph delves into the application of sum and difference formulas for cosine and sine functions. It discusses how to handle cosine and sine expressions involving multiplication and subtraction of angles. The paragraph provides a method to identify the correct formula to use based on the signs and functions present in the expression. It includes examples of finding the value of cosine of 70 degrees times cosine of 40 degrees plus sine of 70 times sine of 40, and cosine of 20 times cosine of 40 minus sine of 20 times sine of 40. The process involves recognizing patterns that correspond to sum and difference formulas and applying them to find the final values.

15:03

📐 Sine and Cosine Values for Specific Angles

The fourth paragraph focuses on finding the values of sine for specific angles using sum and difference formulas. It covers the process of finding sine of 110 degrees multiplied by cosine of 25 degrees plus cosine of 110 times sine of 25, and sine of 5 pi over 12 multiplied by cosine of pi over 12 minus cosine of 5 pi over 12 times sine of pi over 12. The paragraph explains how to identify the correct formula based on the given angles and the signs in the expression. It also shows the reduction of angles to their simplest form and the use of reference angles to find the final sine values.

20:05

🧮 Calculating the Tangent of 15 Degrees

The final paragraph in the script deals with calculating the tangent of 15 degrees. It explains the process of finding two angles that add up to the desired angle and then applying the tangent formula for the difference of angles. The paragraph provides a detailed example of calculating tangent 60 degrees minus tangent 45 degrees over one plus the product of tangent 60 and tangent 45. It shows how to simplify the expression by multiplying by the conjugate and foiling the terms. The final step involves dividing the simplified numerator and denominator to find the value of tangent 15 degrees, which is presented as 2 minus the square root of 3.

Mindmap

Keywords

💡Sum and Difference Formulas

Sum and Difference Formulas are trigonometric identities used to simplify the calculation of the sine, cosine, and tangent of the sum or difference of two angles. These formulas are central to the video's theme, which is to educate on their application in evaluating trigonometric functions without a calculator. For example, the formula for cosine (cos(α ± β) = cos α cos β ∓ sin α sin β) is used to evaluate cos(75 degrees) and cos(5π/12) in the script.

💡Unit Circle

The unit circle is a fundamental concept in trigonometry, defining a circle with a radius of one centered at the origin of a coordinate system. It is crucial for understanding trigonometric functions as angles on this circle correspond to points whose coordinates are the values of the sine and cosine functions. In the script, common angles like 45 degrees and 30 degrees, which are points on the unit circle, are used to derive values for sin 15 degrees and sin 105 degrees using the sum and difference formulas.

💡Exact Values

Exact values in trigonometry refer to the precise trigonometric values derived without approximation, often expressed in radicals instead of decimals. The video script focuses on calculating these exact values using sum and difference formulas, illustrating this with examples like finding the sine of 15 degrees (sin 15° = √6 - √2 / 4) and cosine of 75 degrees.

💡Angle Conversion

Angle conversion between degrees and radians is a necessary skill in trigonometry as different problems and formulas may require different units. The script describes converting 5π/12 radians to degrees to find cosine using sum and difference formulas, emphasizing the practical application of this conversion in solving trigonometric problems.

💡Identity

In trigonometry, an identity is an equation that is true for all values of the variables where both sides of the equation are defined. The video script discusses various trigonometric identities, such as the identity for tangent (tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)), and explains their use in calculating values like tan 15 degrees.

💡Quadrants

Quadrants are the four sections of the coordinate plane, defined by the signs of the x and y coordinates. They are crucial for understanding the signs of trigonometric functions based on the angle's location. The script mentions angles like 120 degrees and 135 degrees, which fall in different quadrants, affecting the signs and values of their trigonometric functions.

💡Simplification

Simplification in the context of trigonometry involves reducing expressions to a more manageable form, often using identities or algebraic manipulations. The video emphasizes this through examples like simplifying the tangent of 15 degrees by using the conjugate to eliminate radicals from the denominator, illustrating the practical applications of algebra in trigonometry.

💡Reference Angle

A reference angle is the acute angle formed by the terminal side of an angle and the horizontal axis. It is used to determine the values of trigonometric functions for angles greater than 90 degrees. In the script, reference angles are used to determine the trigonometric values for angles like 120 degrees and 135 degrees in their respective quadrants.

💡Cosine

Cosine is a trigonometric function that measures the horizontal distance of a point on the unit circle from the origin. It is repeatedly referenced in the video to demonstrate how the sum and difference formulas can be applied, such as in the calculations for cos(75 degrees) and cos(5π/12).

💡Sine

Sine is a trigonometric function that measures the vertical distance of a point on the unit circle from the origin. The script uses it extensively to show the application of sum and difference formulas, notably in calculating sin 15 degrees and sin 105 degrees, providing practical examples of its usage in trigonometry.

Highlights

Sum and difference formulas for cosine, sine, and tangent are introduced.

Cosine of the sum of two angles is equal to the product of cosines minus the product of sines.

Cosine of the difference of two angles is equal to the product of cosines plus the product of sines.

Sine of the sum of two angles is equal to the sum of products of sine and cosine with the opposite sign.

Sine of the difference of two angles is equal to the difference of products of sine and cosine with the same sign.

Tangent of the sum of two angles is given by the formula involving tangents and a fraction with tangents.

Tangent of the difference of two angles is derived similarly to the sum but with a minus sign.

Practical application of these formulas to evaluate sine of 15 degrees using angles 45 and 30.

Exact value of sine 105 degrees is found using angles 60 and 45.

Cosine of an angle expressed as the difference of two angles is evaluated using the sum and difference formulas.

Conversion of radians to degrees to evaluate cosine of 75 degrees using sum and difference formulas.

The expression involving cosine and sine of 70 and 40 degrees is evaluated using the cosine difference formula.

Cosine of 60 degrees is used to find the value of a complex trigonometric expression.

Sine of 110 degrees and cosine of 25 degrees are used to evaluate a trigonometric expression.

The value of tangent 15 degrees is found using angles 60 and 45 with the tangent difference formula.

A method to simplify expressions involving square roots in the denominator is demonstrated.

The final value of tangent 15 degrees is simplified to 2 - square root 3.

Transcripts

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