Name: unit circle trig identities and sum formulas
Uploaded: 2025-11-26T05:44:29.000Z
Duration: 1 min 27 s
Channel: JoVE

A trigonometric identity is an equation involving trigonometric functions that is true for all angles.One such identity, called the Pythagorean identity, can be derived using a unit circle.Any point on this circle corresponds to an angle measured from the positive x-axis.The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.The x and y coordinates form the perpendicular sides of a right triangle, with the radius as the hypotenuse.By the Pythagorean Theorem, the square of the y-coordinate plus the square of the x-coordinate equals the square of the radius.Since the radius is one, this proves that sine squared plus cosine squared equals one.This is known as the Pythagorean identity and holds for all real angles.This identity, derived in two dimensions, has applications that extend to three-dimensional systems. In astronomy, celestial coordinates describe the position of stars on the celestial sphere.The identity ensures that these positions remain consistent on the sphere, allowing astronomers to track stars as they appear to move across the sky.

trigonometric identities i

Trigonometric identities are equations that relate trigonometric functions and hold for all angles within their domains. A fundamental identity among these is the Pythagorean identity, which arises directly from the geometry of the unit circle. For any angle &#952;, a point on the unit circle has coordinates (cos&#8289; &#952;, sin &#8289;&#952;), and since the radius of the circle is one, the Pythagorean Theorem gives:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_1_2__editor_40078.svg' alt='40078_Equation_1_2_.svg' /></figure>This identity serves as the basis for deriving additional identities. Dividing the Pythagorean identity by cos&#8289;2&#952; (assuming cos &#8289;&#952; &#8800; 0) yields:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_2_2__editor_40078.svg' alt='40078_Equation_2_2_.svg' /></figure>The Pythagorean identity also plays a role in more complex identities, such as the cosine addition formula. One method of deriving this formula involves geometric arguments using points on the unit circle and the distance formula. In this derivation, the coordinates of key points involve combinations of sine and cosine values. When applying the distance formula and simplifying, the Pythagorean identity reduces expressions involving the squares of trigonometric terms. This ultimately leads to:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_3_2__editor_40078.svg' alt='40078_Equation_3_2_.svg' /></figure>By replacing t with -t in the addition formula and using the even and odd properties of cosine and sine, the subtraction formula is obtained:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_4_2__editor_40078.svg' alt='40078_Equation_4_2_.svg' /></figure>The sine addition and subtraction formulas can also be derived using cofunction identities and angle transformations:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_5_2__editor_40078.svg' alt='40078_Equation_5_2_.svg' /></figure>Finally, the tangent addition and subtraction identities are derived from the sine and cosine sum formulas by taking their ratio:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_6_editor_40078.svg' alt='40078_Equation_6.svg' /></figure>These identities are essential tools in algebraic simplification, solving trigonometric equations, and calculus applications

Trigonometric Identities I

Trigonometric identities are equations that relate trigonometric functions and hold for all angles within their domains. A fundamental identity among ...

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unit circle trig identities and sum formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity.

That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second Pythagorean identity. This result helps build later trig formulas and shows how one identity can generate others.

The Pythagorean identity also supports the cosine addition formula. One way to derive it uses points on the unit circle and the distance formula. The coordinates of the points involve sine and cosine terms. After simplifying the distance expression, the Pythagorean identity removes square terms and leads to the cosine sum formula.

The subtraction formula for cosine follows from the addition formula. Replacing t with -t and using the even and odd properties of cosine and sine gives the cosine difference identity. The sine addition and subtraction formulas can also be derived through cofunction identities and angle transformations.

The tangent addition and subtraction identities come from the sine and cosine sum formulas. Taking the ratio of the sine and cosine results produces the tangent formulas. These identities are useful for simplifying algebra, solving trigonometric equations, and working in calculus.

Unit Circle Trig Identities and Sum Formulas

unit-circle-trig-identities-and-sum-formulas

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The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements. For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other...

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

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key terms and concepts

Learn how the unit circle leads to Pythagorean, sine, cosine, and tangent identities, including addition and subtraction formulas for high school math.

Unit Circle Trig Identities and Sum Formulas | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Unit Circle Trig Identities and Sum Formulas

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