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Understanding Pythagorean identities

Trigonometry: Master Angles, Functions & Applications

Imaginary and Complex Numbers

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Introduction to imaginary numbers

Complex numbers and complex planes

Adding and subtracting complex numbers

Complex conjugates

Complex Conjugates

Multiplying and dividing complex numbers

Distance and midpoint of complex numbers

Angle and absolute value of complex numbers

Polar form of complex numbers

Operations on complex numbers in polar form

Inverse Trigonometric Functions

Finding inverse trigonometric function from its graph

Evaluating inverse trigonometric functions

Finding inverse reciprocal trigonometric function from its graph

Finding exact value of inverse reciprocal trig functions

Inverse Reciprocal Trigonometric Function: Finding the Exact Value

Inverse reciprocal trigonometric function: finding the exact value

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Trigonometric Ratios and Angle Measure

Unit circle

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

ASTC rule in trigonometry (All Students Take Calculus)

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

Graphing Trigonometric Functions

Sine graph: y = sin x

Cosine graph: y = cos x

Tangent graph: y = tan x

Cotangent graph: y = cot x

Secant graph: y = sec x

Cosecant graph: y = csc x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Applications of Trigonometric Functions

Ferris wheel trig problems

Tides and water depth trig problems

Spring (simple harmonic motion) trig problems

Trigonometric Identities

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Sum and Difference Identities

Double-angle identities

Cofunction identities

Solving Trigonometric Equations

Solving first degree trigonometric equations

Solving second degree trigonometric equations

Solving trigonometric equations involving multiple angles

Determining non-permissible values for trig expressions

Solving trigonometric equations using pythagorean identities

Solving trigonometric equations using Pythagorean identities

Solving trigonometric equations using sum and difference identities

Solving trigonometric equations using double-angle identities

Right Triangle Trigonometry

Use sine ratio to calculate angles and sides (Sin = \(  \frac{o}{h}\) )

Use cosine ratio to calculate angles and sides (Cos =  \(  \frac{a}{h}\)  )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles 

Use tangent ratio to calculate angles and sides (Tan =  \(  \frac{o}{a}\)  )

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Ex. 1:

Proving all three Pythagorean identities using the unit circle

Ex. 2:

Ex. 3

Simplifying secant squared minus one times cotangent squared

Proving sinx/(1+cosx) + sinx/(1-cosx) = 2cscx

Simplifying a trig expression using conjugates and Pythagorean identities

Proving tanx(cscx+1) equals cotx/(cscx-1) using conjugates

Pythagorean identities are formulas, derived from Pythagorean Theorem, that allow us to find out where a point is on the unit circle. Learn the tricks and tips on how to use the unit circle to derive and prove the Pythagorean identities can be difficult. 

Factoring difference of squares: x^2 - y^2

Pythagorean identities

What You'll Learn

Derive the three Pythagorean identities from the unit circle and Pythagorean theorem

Apply sine²θ + cos²θ = 1 to prove tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ

Recognize when to substitute Pythagorean identities in trigonometric expressions

Use quotient and reciprocal identities to convert between trig functions

Simplify complex expressions by combining fractions and applying conjugates

What You'll Practice

Proving Pythagorean identities using the unit circle definition

Simplifying expressions with sec²x - 1 and other Pythagorean forms

Combining trigonometric fractions with common denominators

Multiplying by conjugates to create difference of squares patterns

Verifying complex trigonometric identities step-by-step

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

5 Video lessons

Practice exercises

Learning resources

Skills

Pythagorean Identities

Unit Circle

Trigonometry

Identity Proofs

Algebraic Manipulation

Conjugates

Simplification