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Trigonometric Identities

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
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- Derivatives
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- Derivatives of Inverse Trigonometric Functions
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- Determining Volumes by Slicing
- Direction Fields
- Disk Method
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- Eliminating the Parameter
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- Evaluating a Definite Integral
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- Implicit Differentiation Tangent Line
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- Composition
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- Fundamentals of Geometry
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- HL ASA and AAS
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
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- Plane Geometry
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- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
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- Rotations
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- Similarity
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- Mechanics Maths
- Acceleration and Time
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- Assumptions
- Calculus Kinematics
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- Kinematics
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- Power
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- Probability and Statistics
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- Independent Events Probability
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- Pure Maths
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- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
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- Approximation and Estimation
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- Determinants
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- Equations
- Equations and Identities
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- Estimation in Real Life
- Euclidean Algorithm
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- Even Functions
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- Expression Math
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- Faces Edges and Vertices
- Factorials
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- Finding Maxima and Minima Using Derivatives
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- Forms of Quadratic Functions
- Fractional Powers
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- Fractions
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- Function Basics
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- Functions
- Fundamental Counting Principle
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- Generating Terms of a Sequence
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- Greatest Common Divisor
- Growth and Decay
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- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
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- Inequalities Maths
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- Integers
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- Integration
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- Interest
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- Inverse Matrices
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- Iterative Methods
- L'Hopital's Rule
- Law of Cosines in Algebra
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- Laws of Logs
- Limits of Accuracy
- Linear Expressions
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- Math formula
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- Notation
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- Problem-solving Models and Strategies
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- Proof
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- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
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- Scale Drawings and Maps
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- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
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- Simplifying Fractions
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- Simultaneous Equations
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- Small Angle Approximation
- Solving Linear Equations
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- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
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- Solving and Graphing Quadratic Equations
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- Special Products
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- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
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- Surds
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- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
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- Bivariate Data
- Box Plots
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- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
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- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
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- Conducting a Study
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- Confidence Interval for Population Mean
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- Confidence Interval for Slope of Regression Line
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- Confidence Intervals
- Correlation Math
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- Data Analysis
- Data Interpretation
- Degrees of Freedom
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- Generalization and Conclusions
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- Inference for Distributions of Categorical Data
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Jetzt kostenlos anmeldenTrigonometric identities are important to work through a variety of problems and advanced equations. They allow us to simplify many problems and make situations easier.

There are two main formulaic identities that must be learnt to prove and solve other equations. These are:

and $\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=\mathrm{tan}x$

Let’s prove these identities starting with ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$.

PROOF:

Firstly let’s draw a triangle with angle θ.

General Triangle of angle θ

Now if we write out expressions for a and b using SOHCAHTOA we get:$a=c\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}b=c\mathrm{cos}\theta $Therefore:

$\frac{a}{c}=\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}\frac{b}{c}=\mathrm{cos}\theta $Now if we square both of these expressions for sin and cos we get:

$\frac{{a}^{2}}{{c}^{2}}={\mathrm{sin}}^{2}\theta \phantom{\rule{0ex}{0ex}}\frac{{b}^{2}}{{c}^{2}}={\mathrm{cos}}^{2}\theta $Summing these we get:

${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =\frac{{a}^{2}+{b}^{2}}{{c}^{2}}$By Pythagoras' theorem:

${a}^{2}+{b}^{2}={c}^{2}\phantom{\rule{0ex}{0ex}}$Therefore:

$\frac{{a}^{2}+{b}^{2}}{{c}^{2}}=\frac{{c}^{2}}{{c}^{2}}=1\phantom{\rule{0ex}{0ex}}{\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$Now let’s move on to proving $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$. The first half of this proof is identical to the proof above.

PROOF:

Firstly let’s draw a triangle with angle θ.

Now if we write out expressions for a and b using SOHCAHTOA we get:So Now if we divide these two expressions for sin and cos:$\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=\frac{\left(\frac{a}{c}\right)}{\left(\frac{b}{c}\right)}$$=\frac{a}{c}\times \frac{c}{b}=\frac{a}{b}$This is an expression for the opposite side over the adjacent side, therefore:

$\frac{a}{b}=\mathrm{tan}\theta $Therefore:

$\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=\mathrm{tan}\theta $Now let’s look at some worked examples where trigonometric identities can be applied.

Solve the equation $4{\mathrm{sin}}^{2}x+8\mathrm{cos}x-7=0$ for $0\le x\le 180.$

SOLUTION:The first thing to do would be to substitute$1-{\mathrm{cos}}^{2}x$for ${\mathrm{sin}}^{2}x$ .The equation now ends up being $4(1-{\mathrm{cos}}^{2}x)+8\mathrm{cos}x-7=0$ .Simplifying this further:$4-4{\mathrm{cos}}^{2}x+8\mathrm{cos}x-7=0$$4{\mathrm{cos}}^{2}x-8\mathrm{cos}x+3=0$Now we can solve this like a quadratic by taking $y=\mathrm{cos}x$.$4{y}^{2}-8y+3=0\phantom{\rule{0ex}{0ex}}(2y-1)(2y-3)=0\phantom{\rule{0ex}{0ex}}y=0.5ory=1.5$Now we need to do x = cosLet's look at another example of rearranging trigonometric identities.

Show that the equation $2\mathrm{sin}x=\frac{(4\mathrm{cos}x-1)}{\mathrm{tan}x}$ can be written as $6{\mathrm{cos}}^{2}x-\mathrm{cos}x-2=0.$

SOLUTION:Firstly let’s rearrange to get rid of any denominators.$2\mathrm{sin}x\mathrm{tan}x=4\mathrm{cos}x-1$Now let’s replace $\mathrm{tan}x$ with $\frac{\mathrm{sin}x}{\mathrm{cos}x}$:$2\mathrm{sin}x\frac{\mathrm{sin}x}{\mathrm{cos}x}=4\mathrm{cos}x-1$$\frac{2{\mathrm{sin}}^{2}x}{\mathrm{cos}x}=4\mathrm{cos}x-1$Now get rid of the denominator by multiplying through by $\mathrm{cos}x$:$2{\mathrm{sin}}^{2}x=4{\mathrm{cos}}^{2}x-\mathrm{cos}x$Now replace ${\mathrm{sin}}^{2}x$ with $1-{\mathrm{cos}}^{2}x$:$2(1-{\mathrm{cos}}^{2}x)=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}2-2{\mathrm{cos}}^{2}x=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}$Now rearrange this equation:$2=6{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}6{\mathrm{cos}}^{2}x-\mathrm{cos}x-2=0$QEDFirstly we need to know three new bits of terminology:

$secx=\frac{1}{\mathrm{cos}x}\phantom{\rule{0ex}{0ex}}\mathrm{cos}ecx=\frac{1}{\mathrm{sin}x}\phantom{\rule{0ex}{0ex}}cotx=\frac{1}{\mathrm{tan}x}$

These are all reciprocals of standard sin, cos and tan.

Now let’s look at the identity ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$:

If we divide the entire equation by ${\mathrm{cos}}^{2}\left(x\right)$we get:$\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{cos}}^{2}x}=\frac{1}{{\mathrm{cos}}^{2}x}$Now using the identity $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$:${\mathrm{tan}}^{2}x+1=se{c}^{2}x$This is our first new identity. Now if we divide our entire equation by ${\mathrm{sin}}^{2}x$$\frac{{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x}=\frac{1}{{\mathrm{sin}}^{2}x}$Now using the identity $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$, so :$1+\frac{1}{{\mathrm{tan}}^{2}x}=\frac{1}{{\mathrm{sin}}^{2}x}\phantom{\rule{0ex}{0ex}}1+co{t}^{2}x=\mathrm{cos}e{c}^{2}x$Now we have our two new identities:${\mathrm{tan}}^{2}x+1=se{c}^{2}x\phantom{\rule{0ex}{0ex}}co{t}^{2}x+1=\mathrm{cos}e{c}^{2}x$Let’s see them in action in some worked examples.

Solve, for 0 ≤ θ < 360°, the equation:

$2{\mathrm{tan}}^{2}x+secx=1$to 1 dp.We can see that if we perform the identity $\mathrm{cos}x=\mathrm{cos}(360-x)$, the other value of $x$ is $360-131.8=228.2$.

Then we need to perform ${\mathrm{cos}}^{-1}\left(1\right)=0$, again using the identity $\mathrm{cos}x=\mathrm{cos}(360-x)$, $x=360$.

So to 1 decimal place our 4 solutions in degrees are:

$x=131.8,x=228.2,x=0,x=360$

Trigonometric identities are used to derive new formulae and equations.

They can help solve equations involving trigonometry.

They help us geometrically visualise real-life situations.

They have proofs, which can be adapted from basic trigonometry.

Images:

Graph of y=cos x: https://commons.wikimedia.org/wiki/File:Cos(x).PNG

sinx/cosx=tanx, sin^2(x)+cos^2(x)=1. 1/cosx=secx

Simply rearrange to the identities listed above and substitute them back in.

Drawing a diagram reveals why each identity works. Regular SOHCAHTOA can show what’s going on.

They can help us solve larger trigonometric equations that cannot be solved otherwise.

More about Trigonometric Identities

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