Trigonometry Guide
Learn the essential trigonometric identities and how to use them to rewrite, simplify, and verify trigonometric expressions.
Trigonometric Relationships
A trigonometric identity is an equation involving trig functions that is true for every value for which both sides of the equation are defined.
Instead of treating identities as a long list of formulas to memorize, it is more useful to understand how the identities are connected and how one form can be rewritten as another.
Identity
An identity is not an equation that is true for only one particular value of \( \theta \). It represents a relationship that remains true throughout its domain.
Equation
This is true only for certain values of \( \theta \).
Identity
This relationship is true for every angle where the expressions are defined.
Why This Matters
The same trigonometric quantity can often be written in several different ways. That means a complicated-looking expression may become much easier once one part is replaced using an identity.
Learning trig identities is therefore less about memorizing isolated equations and more about recognizing equivalent forms.
The Essential Formulas
Most introductory trig identity problems rely heavily on three groups: reciprocal identities, quotient identities, and Pythagorean identities.
Family 1
The reciprocal identities come from the fact that cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent.
Cosecant and Sine
Secant and Cosine
Cotangent and Tangent
Think “Flip”
Reciprocal identities are especially useful when you want to replace \(\sec\), \(\csc\), or \(\cot\) with expressions involving sine, cosine, or tangent.
Family 2
Tangent and cotangent can also be written as ratios involving sine and cosine.
Tangent
Tangent is sine divided by cosine.
Cotangent
Cotangent is cosine divided by sine.
Why This Is Useful
When a trig expression looks complicated, one of the most useful strategies is to rewrite tangent, cotangent, secant, and cosecant using sine and cosine.
Watch the Order
Tangent is \(\sin\theta/\cos\theta\), while cotangent is the reverse: \(\cos\theta/\sin\theta\).
Family 3
The Pythagorean identities are some of the most important identities in trigonometry.
The key is that you do not really need to think of them as three unrelated formulas. Start with one fundamental identity, and the other two can be derived from it.
Fundamental Identity
Tangent-Secant Identity
Cotangent-Cosecant Identity
Where Does It Come From?
Recall that every point on the unit circle can be written as
The equation of a circle centered at the origin with radius \(1\) is
On the unit circle,
Substitute these into the equation of the circle.
Notation Reminder
\[ \sin^2\theta \] means \[ (\sin\theta)^2 \] It does not mean \[ \sin(\theta^2) \]
Derivation 1
Start with the fundamental Pythagorean identity.
Divide every term by \(\cos^2\theta\).
Now use the quotient and reciprocal identities.
Derivation 2
Again, begin with
This time, divide every term by \(\sin^2\theta\).
Simplify each term.
See the Connection
If you forget one of the last two identities, you can reconstruct it from \[ \sin^2\theta+\cos^2\theta=1. \]
Family 4
Sum and difference identities let you find trig values for angles that can be written as the sum or difference of two familiar angles.
They also form the foundation for several other identity families, including the double-angle formulas.
Sine of a Sum
Sine of a Difference
Sine Pattern
Cosine of a Sum
Cosine of a Difference
Cosine Pattern
Tangent of a Sum
Tangent of a Difference
Watch the Signs
Sine keeps the same sign as the angle operation. Cosine switches the sign. Tangent keeps the sign in the numerator and switches it in the denominator.
Example
Use the sine sum identity.
\[ \sin(A+B) = \sin A\cos B + \cos A\sin B \]Substitute \(A=45^\circ\) and \(B=30^\circ\).
\[ \sin75^\circ = \sin45^\circ\cos30^\circ + \cos45^\circ\sin30^\circ \]Substitute the exact trig values.
\[ = \left( \frac{\sqrt2}{2} \right) \left( \frac{\sqrt3}{2} \right) + \left( \frac{\sqrt2}{2} \right) \left( \frac12 \right) \]Simplify.
\[ \boxed{ \sin75^\circ = \frac{ \sqrt6+\sqrt2 }{ 4 } } \]Family 5
Double-angle identities are special cases of the sum identities where the two angles are the same.
In other words, replace \(A\) and \(B\) with \(\theta\).
From the Sine Sum Identity
Let \(A=B=\theta\).
From the Cosine Sum Identity
Again, let \(A=B=\theta\).
Cosine Has Three Useful Forms
The Pythagorean identity lets us rewrite the cosine double-angle formula in two additional ways.
Form 1
Form 2
Form 3
From the Tangent Sum Identity
Let \(A=B=\theta\).
Double-Angle Reference
Family 6
Half-angle identities let you work with trig functions of \(\theta/2\).
They come from rearranging the cosine double-angle identities.
From the Cosine Double-Angle Identity
Solve for \(\sin^2\alpha\).
Let \(2\alpha=\theta\), so \(\alpha=\theta/2\).
From the Cosine Double-Angle Identity
Solve for \(\cos^2\alpha\).
Again, let \(2\alpha=\theta\).
What Determines the \(\pm\)?
The sign is determined by the quadrant containing \(\theta/2\). Use the sign that matches the trig function in that quadrant.
Tangent Half Angle
Square-Root Form
Sine Over One Plus Cosine
One Minus Cosine Over Sine
Example
Use the cosine half-angle identity.
\[ \cos\left( \frac{\theta}{2} \right) = \pm \sqrt{ \frac{ 1+\cos\theta }{ 2 } } \]Use \(\theta=30^\circ\). Since \(15^\circ\) is in Quadrant I, cosine is positive.
\[ \cos15^\circ = \sqrt{ \frac{ 1+\cos30^\circ }{ 2 } } \]Substitute \(\cos30^\circ = \sqrt3/2\).
\[ \cos15^\circ = \sqrt{ \frac{ 1+\frac{\sqrt3}{2} }{ 2 } } \]Simplify.
\[ \boxed{ \cos15^\circ = \frac{ \sqrt{ 2+\sqrt3 } }{ 2 } } \]Family 7
Power-reduction identities rewrite squared trig functions using first powers of sine or cosine.
They come directly from the cosine double-angle identities.
Sine Squared
Cosine Squared
Tangent Squared
Where They Come From
Why “Power Reduction”?
These identities reduce a squared trig function to an expression involving a first-power cosine term. They become especially useful later in calculus when working with trig integrals.
Put Them to Work
Once you know the core identities, the next step is learning how to rearrange and combine them.
Strategy 1
The Pythagorean identities can be rewritten to isolate whichever trig expression you need.
From
Or
From
From
Look for a Matching Piece
If an expression contains \(1-\cos^2\theta\), think \(\sin^2\theta\). If it contains \(\sec^2\theta-1\), think \(\tan^2\theta\).
Strategy 2
A useful identity can turn a complicated expression into something much simpler.
Example 1
Recognize the Pythagorean form:
\[ 1-\cos^2\theta = \sin^2\theta \]Substitute.
\[ \frac{ \sin^2\theta }{ \sin\theta } \]Cancel one factor of \(\sin\theta\).
\[ \boxed{ \sin\theta } \]Example 2
Use the rearranged identity:
\[ \sec^2\theta-1 = \tan^2\theta \]Substitute.
\[ \frac{ \tan^2\theta }{ \tan\theta } \]Simplify.
\[ \boxed{ \tan\theta } \]Strategy 3
When you are unsure what to do, rewriting everything in terms of sine and cosine is often a strong first move.
Example 3
Rewrite tangent and secant.
\[ \tan\theta = \frac{ \sin\theta }{ \cos\theta } \] \[ \sec\theta = \frac{ 1 }{ \cos\theta } \]Substitute.
\[ \frac{ \frac{ \sin\theta }{ \cos\theta } }{ \frac{ 1 }{ \cos\theta } } \]Multiply by the reciprocal.
\[ \frac{ \sin\theta }{ \cos\theta } \cdot \cos\theta \]Cancel \(\cos\theta\).
\[ \boxed{ \sin\theta } \]Strategy Check
Consider a reciprocal identity.
Consider rewriting with sine and cosine.
Look for a Pythagorean identity.
Rewrite everything in sine and cosine and simplify algebraically.
Proving Relationships
To verify an identity, you show that one side can be transformed into the other using valid identities and algebra.
The Main Rule
Start with the more complicated side, rewrite and simplify it, and stop when it matches the other side.
Good Approach
Avoid This
If you manipulate both sides simultaneously, it becomes much harder to show that one side truly transforms into the other.
A Reliable Process
Start where there is more to simplify.
Check for reciprocal, quotient, and Pythagorean forms.
Factor, combine fractions, cancel common factors, or simplify powers.
Once your transformed side matches the untouched side, the identity is verified.
Verification Example 1
Work only on the left side.
\[ \frac{ 1-\cos^2\theta }{ \sin\theta } \]Use \(1-\cos^2\theta = \sin^2\theta\).
\[ \frac{ \sin^2\theta }{ \sin\theta } \]Simplify.
\[ \sin\theta \]Verification Example 2
Begin with the left side.
\[ \tan\theta\cos\theta \]Rewrite tangent using the quotient identity.
\[ \frac{ \sin\theta }{ \cos\theta } \cos\theta \]Cancel \(\cos\theta\).
\[ \sin\theta \]Verification Example 3
Start with the left side.
\[ \frac{ \sec^2\theta-1 }{ \tan\theta } \]Use the Pythagorean identity \(\sec^2\theta-1 = \tan^2\theta\).
\[ \frac{ \tan^2\theta }{ \tan\theta } \]Simplify.
\[ \tan\theta \]If You Get Stuck
Try rewriting everything in sine and cosine. It is not always the shortest method, but it often reveals factors that cancel or combine.
Watch Out
Most identity mistakes come from algebra, notation, or using the correct identity in the wrong way.
Remember:
The expression \(\sin^2\theta\) means \((\sin\theta)^2\), not \(\sin(\theta^2)\).
When dividing a Pythagorean identity by \(\sin^2\theta\) or \(\cos^2\theta\), divide every term in the equation.
You can cancel common factors, but not individual pieces of terms that are being added or subtracted.
When verifying an identity, simplify one side until it becomes the other side.
If an identity does not immediately help, try another form or rewrite the expression in sine and cosine.
Quick Reference
Reciprocal
Quotient
Pythagorean
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