9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
3 min readโขjune 24, 2024
Trigonometric identities are essential equations that hold true for all angles. They're like the building blocks of trigonometry, helping us simplify complex expressions and solve tricky problems. Knowing these identities is crucial for tackling more advanced math.
We'll look at different types of identities, including Pythagorean, reciprocal, and quotient. We'll also explore how to verify and simplify trigonometric expressions using these identities and other algebraic techniques. This knowledge is key for mastering trigonometry.
Verifying Trigonometric Identities
Application of fundamental trigonometric identities
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Trigonometric identity represents an equation true for all values of the variable where both sides are defined (sin2ฮธ+cos2ฮธ=1 holds for all ฮธ)
Pythagorean identities express relationships between trigonometric functions
Factor out common terms (2sinฮธ+4cosฮธ=2(sinฮธ+2cosฮธ))
Combine like terms (3sin2ฮธ+5sin2ฮธ=8sin2ฮธ)
Multiply by a form of 1 (sinฮธsinฮธโ or cosฮธcosฮธโ) to facilitate simplification
Pythagorean identities enable substitution and simplification
Replace sin2ฮธ with 1โcos2ฮธ or vice versa (1โsin2ฮธ=cos2ฮธ)
Replace tan2ฮธ with sec2ฮธโ1 or vice versa (1+tan2ฮธ=sec2ฮธ)
Replace cot2ฮธ with csc2ฮธโ1 or vice versa (1+cot2ฮธ=csc2ฮธ)
Reciprocal and quotient identities allow rewriting expressions
Replace cscฮธ with sinฮธ1โ or vice versa (sinฮธ=cscฮธ1โ)
Replace secฮธ with cosฮธ1โ or vice versa (cosฮธ=secฮธ1โ)
Replace cotฮธ with sinฮธcosฮธโ or vice versa (tanฮธ=cosฮธsinฮธโ)
Algebraic manipulation is crucial in simplifying complex trigonometric expressions
Patterns in trigonometric transformations
Common trigonometric patterns have equivalent forms
acosฮธ+bsinฮธ=a2+b2โcos(ฮธโarctan(abโ)) (sum of cosine and sine)
acosฮธโbsinฮธ=a2+b2โcos(ฮธ+arctan(abโ)) (difference of cosine and sine)
Double-angle formulas express trigonometric functions of double angles
sin(2ฮธ)=2sinฮธcosฮธ (sine of double angle)
cos(2ฮธ)=cos2ฮธโsin2ฮธ=2cos2ฮธโ1=1โ2sin2ฮธ (cosine of double angle)
Half-angle formulas relate trigonometric functions of half angles to cosine
sin2(2ฮธโ)=21โcosฮธโ (sine squared of half angle)
cos2(2ฮธโ)=21+cosฮธโ (cosine squared of half angle)
Sum and difference formulas expand or simplify expressions involving sums or differences of angles
sin(ฮฑยฑฮฒ)=sinฮฑcosฮฒยฑcosฮฑsinฮฒ (sine of sum or difference)
cos(ฮฑยฑฮฒ)=cosฮฑcosฮฒโsinฮฑsinฮฒ (cosine of sum or difference)
Fundamental Concepts in Trigonometry
Fundamental trigonometric functions (sine, cosine, tangent) form the basis for all trigonometric identities and expressions
The unit circle provides a visual representation of trigonometric functions and their relationships
Radian measure is often used in trigonometric equations and identities, allowing for more concise expressions
Trigonometric equations involve solving for unknown angles or variables using trigonometric functions and identities
Key Terms to Review (6)
Conic: A conic is a curve obtained by intersecting a plane with a double-napped cone. The types of conics include ellipses, hyperbolas, and parabolas.
Ellipse: An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (foci) is constant. It is an important type of conic section.
Major and minor axes: The major axis of an ellipse is the longest diameter, passing through the center and both foci. The minor axis is perpendicular to the major axis at the center, representing the shortest diameter.
Translation: Translation is a type of transformation that shifts every point of a shape or graph a constant distance in a specified direction. In the context of ellipses, it involves moving the entire ellipse without changing its shape, size, or orientation.
Vertex: The vertex is the highest or lowest point on the graph of a quadratic function. It represents the maximum or minimum value of the function.
Vertices: Vertices are the points on an ellipse that lie on the major axis and are at the maximum distance from the center. They are crucial for determining the shape and size of the ellipse.
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