The Central Idea

For an acute angle \(\theta\) in a right triangle, the three main trigonometric functions are

\[ \boxed{ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan\theta=\frac{\text{opposite}}{\text{adjacent}} }. \]

Each function compares a different pair of sides. Together, they describe the shape of a right triangle relative to the chosen angle.

Three Functions, One Angle

Suppose \(\theta\) is one acute angle of a right triangle.

The same angle determines all three ratios:

These interpretations become especially important later when trigonometry moves from triangles to coordinates, vectors, circles, and waves.

Sine

The sine of \(\theta\) is

\[ \boxed{\sin\theta=\frac{O}{H}}. \]

The numerator is the side opposite the angle, and the denominator is the hypotenuse.

In a right triangle, the opposite side is always shorter than the hypotenuse, so for an acute angle,

\[ 0<\sin\theta<1. \]

Cosine

The cosine of \(\theta\) is

\[ \boxed{\cos\theta=\frac{A}{H}}. \]

The numerator is the side adjacent to the angle, and the denominator is the hypotenuse.

Since the adjacent leg is also shorter than the hypotenuse,

\[ 0<\cos\theta<1 \]

for an acute angle.

Tangent

The tangent of \(\theta\) is

\[ \boxed{\tan\theta=\frac{O}{A}}. \]

Tangent compares the two legs rather than comparing a leg with the hypotenuse.

Because either leg may be longer, tangent can be less than \(1\), equal to \(1\), or greater than \(1\).

SOH-CAH-TOA

The familiar mnemonic SOH-CAH-TOA summarizes the three definitions:

\[ \boxed{ \begin{aligned} \text{SOH:}\quad &\sin\theta=\frac{O}{H},\\ \text{CAH:}\quad &\cos\theta=\frac{A}{H},\\ \text{TOA:}\quad &\tan\theta=\frac{O}{A}. \end{aligned} } \]

The mnemonic is useful for recall, but understanding the side roles is more important than memorizing the letters.

A \(3\)-\(4\)-\(5\) Example

Consider a right triangle with side lengths

\[ 3,\qquad4,\qquad5. \]

Let \(\theta\) be the acute angle opposite the side of length \(3\).

Then

\[ \boxed{\sin\theta=\frac35}, \] \[ \boxed{\cos\theta=\frac45}, \]

and

\[ \boxed{\tan\theta=\frac34}. \]

The Same Triangle from the Other Angle

Let \(\phi\) be the other acute angle in the same \(3\)-\(4\)-\(5\) triangle.

The side of length \(4\) is opposite \(\phi\), and the side of length \(3\) is adjacent to \(\phi\).

Therefore,

\[ \sin\phi=\frac45, \qquad \cos\phi=\frac35, \qquad \tan\phi=\frac43. \]

Changing the reference angle changes the side roles and therefore changes the ratios.

Sine and Cosine as Complementary Partners

The acute angles of a right triangle satisfy

\[ \phi=90^\circ-\theta. \]

Because opposite and adjacent switch when the reference angle changes,

\[ \boxed{\sin\theta=\cos(90^\circ-\theta)} \]

and

\[ \boxed{\cos\theta=\sin(90^\circ-\theta)}. \]

Sine and cosine are therefore closely linked rather than independent functions.

Tangent from Sine and Cosine

Since

\[ \sin\theta=\frac{O}{H} \qquad\text{and}\qquad \cos\theta=\frac{A}{H}, \]

their quotient is

\[ \frac{\sin\theta}{\cos\theta} = \frac{O/H}{A/H} = \frac{O}{A}. \]

Therefore,

\[ \boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}. \]

The Pythagorean Identity Begins in a Triangle

Let the opposite side be \(O\), the adjacent side be \(A\), and the hypotenuse be \(H\).

The Pythagorean relationship gives

\[ O^2+A^2=H^2. \]

Divide every term by \(H^2\):

\[ \frac{O^2}{H^2}+\frac{A^2}{H^2}=1. \]

Since

\[ \frac{O}{H}=\sin\theta \qquad\text{and}\qquad \frac{A}{H}=\cos\theta, \]

we obtain

\[ \boxed{\sin^2\theta+\cos^2\theta=1}. \]

This identity will become fundamental throughout trigonometry.

What \(\sin^2\theta\) Means

The notation

\[ \sin^2\theta \]

means

\[ (\sin\theta)^2. \]

It does not mean

\[ \sin(\theta^2). \]

Similarly,

\[ \cos^2\theta=(\cos\theta)^2. \]

Finding One Function from Another

Suppose \(\theta\) is acute and

\[ \sin\theta=\frac{5}{13}. \]

Think of a right triangle with opposite side \(5\) and hypotenuse \(13\).

The adjacent side is

\[ A=\sqrt{13^2-5^2}=\sqrt{144}=12. \]

Therefore,

\[ \boxed{\cos\theta=\frac{12}{13}} \]

and

\[ \boxed{\tan\theta=\frac{5}{12}}. \]

Using the Pythagorean Identity Directly

If

\[ \sin\theta=\frac{5}{13}, \]

then

\[ \sin^2\theta+\cos^2\theta=1 \]

gives

\[ \left(\frac{5}{13}\right)^2+\cos^2\theta=1. \]

Thus,

\[ \cos^2\theta = 1-\frac{25}{169} = \frac{144}{169}. \]

Since \(\theta\) is acute, cosine is positive:

\[ \boxed{\cos\theta=\frac{12}{13}}. \]

Why Sine and Cosine Cannot Exceed \(1\) in a Right Triangle

Both sine and cosine divide a leg by the hypotenuse:

\[ \sin\theta=\frac{O}{H}, \qquad \cos\theta=\frac{A}{H}. \]

The hypotenuse is the longest side, so

\[ O Therefore both ratios lie strictly between \(0\) and \(1\) for acute angles.

Why Tangent Can Exceed \(1\)

Tangent compares the legs:

\[ \tan\theta=\frac{O}{A}. \]

If the opposite leg is shorter than the adjacent leg, then

\[ \tan\theta<1. \]

If the legs are equal,

\[ \tan\theta=1. \]

If the opposite leg is longer,

\[ \tan\theta>1. \]

Tangent and Steepness

Tangent naturally measures steepness because

\[ \tan\theta=\frac{\text{rise}}{\text{run}}. \]

In coordinate geometry, slope is

\[ m=\frac{\text{rise}}{\text{run}}. \]

Therefore, when a line makes an acute angle \(\theta\) with the horizontal,

\[ \boxed{m=\tan\theta}. \]

A larger tangent corresponds to a steeper positive line.

Angles Smaller than \(45^\circ\)

If an acute angle satisfies

\[ 0^\circ<\theta<45^\circ, \]

then the opposite leg is shorter than the adjacent leg.

Therefore,

\[ \tan\theta<1. \]

In this range, cosine is greater than sine:

\[ \cos\theta>\sin\theta. \]

At \(45^\circ\)

A \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle has equal legs.

Therefore,

\[ \tan45^\circ=1. \]

Also,

\[ \sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2}. \]

The angle \(45^\circ\) is the acute angle where sine and cosine are equal.

Angles Larger than \(45^\circ\)

If

\[ 45^\circ<\theta<90^\circ, \]

then the opposite leg is longer than the adjacent leg.

Therefore,

\[ \tan\theta>1 \]

and

\[ \sin\theta>\cos\theta. \]

How Sine Changes Across Acute Angles

As an acute angle increases from near \(0^\circ\) toward \(90^\circ\), the opposite side becomes larger relative to the hypotenuse.

Therefore, sine increases:

\[ \sin\theta \]

moves from values near \(0\) toward values near \(1\).

A later lesson will extend this behavior into a full trigonometric graph.

How Cosine Changes Across Acute Angles

As an acute angle increases toward \(90^\circ\), the adjacent side becomes smaller relative to the hypotenuse.

Therefore, cosine decreases:

\[ \cos\theta \]

moves from values near \(1\) toward values near \(0\).

How Tangent Changes Across Acute Angles

For a very small acute angle, the opposite leg is small compared with the adjacent leg, so tangent is near \(0\).

As the angle approaches \(90^\circ\), the adjacent leg becomes very small relative to the opposite leg.

Thus tangent grows very large.

This behavior anticipates the vertical asymptotes that appear in the graph of the tangent function.

Special Angle Values

Several exact values recur throughout trigonometry:

\[ \begin{array}{c|ccc} \theta & \sin\theta & \cos\theta & \tan\theta\\ \hline 30^\circ & \frac12 & \frac{\sqrt3}{2} & \frac{\sqrt3}{3}\\[4pt] 45^\circ & \frac{\sqrt2}{2} & \frac{\sqrt2}{2} & 1\\[4pt] 60^\circ & \frac{\sqrt3}{2} & \frac12 & \sqrt3 \end{array} \]

These values come directly from the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) and \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangles.

Deriving the \(45^\circ\) Values

Start with a right triangle whose legs both have length \(1\).

The hypotenuse is

\[ \sqrt{1^2+1^2}=\sqrt2. \]

Therefore,

\[ \sin45^\circ=\frac{1}{\sqrt2} =\frac{\sqrt2}{2}, \] \[ \cos45^\circ=\frac{1}{\sqrt2} =\frac{\sqrt2}{2}, \]

and

\[ \tan45^\circ=\frac11=1. \]

Deriving the \(30^\circ\) and \(60^\circ\) Values

A \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle has side ratio

\[ 1:\sqrt3:2. \]

Relative to the \(30^\circ\) angle:

\[ O=1,\qquad A=\sqrt3,\qquad H=2. \]

Therefore,

\[ \sin30^\circ=\frac12, \] \[ \cos30^\circ=\frac{\sqrt3}{2}, \]

and

\[ \tan30^\circ=\frac{1}{\sqrt3} =\frac{\sqrt3}{3}. \]

The \(60^\circ\) Values from Complementarity

Since

\[ 60^\circ=90^\circ-30^\circ, \]

the complementary relationships give

\[ \sin60^\circ=\cos30^\circ=\frac{\sqrt3}{2} \]

and

\[ \cos60^\circ=\sin30^\circ=\frac12. \]

Then

\[ \tan60^\circ = \frac{\sin60^\circ}{\cos60^\circ} = \frac{\sqrt3/2}{1/2} = \sqrt3. \]

Exact Form versus Decimal Approximation

An expression such as

\[ \frac{\sqrt3}{2} \]

is exact.

Its decimal approximation is

\[ 0.866025\ldots \]

Exact forms preserve complete mathematical information and should usually be retained unless a decimal value is required.

Evaluating with a Calculator

For an angle without a standard exact value, a calculator provides a numerical approximation.

For example,

\[ \sin37^\circ\approx0.6018, \] \[ \cos37^\circ\approx0.7986, \]

and

\[ \tan37^\circ\approx0.7536. \]

These three numbers describe the same angle in three different ways.

Degree Mode Matters

If the input contains a degree symbol, such as

\[ 37^\circ, \]

the calculator must interpret the number as degrees.

Entering \(37\) in radian mode evaluates a different angle and produces different values.

Recovering an Angle with Inverse Functions

If a trigonometric value is known, an inverse trigonometric function can recover the corresponding acute angle.

For example, if

\[ \sin\theta=0.6, \]

then

\[ \theta=\sin^{-1}(0.6)\approx36.87^\circ. \]

Similarly,

\[ \theta=\cos^{-1}(0.8)\approx36.87^\circ \]

and

\[ \theta=\tan^{-1}(0.75)\approx36.87^\circ. \]

Three Different Ratios Can Identify the Same Angle

In a \(3\)-\(4\)-\(5\) triangle,

\[ \sin\theta=\frac35, \qquad \cos\theta=\frac45, \qquad \tan\theta=\frac34. \]

Therefore,

\[ \sin^{-1}\left(\frac35\right), \qquad \cos^{-1}\left(\frac45\right), \qquad \tan^{-1}\left(\frac34\right) \]

all produce the same acute angle.

Sine as a Vertical Fraction

If a segment of length \(r\) makes an acute angle \(\theta\) with the horizontal, its vertical component is

\[ y=r\sin\theta. \]

Rearranging,

\[ \sin\theta=\frac{y}{r}. \]

This is the same opposite-over-hypotenuse ratio expressed in coordinate language.

Cosine as a Horizontal Fraction

The horizontal component of the same segment is

\[ x=r\cos\theta. \]

Therefore,

\[ \cos\theta=\frac{x}{r}. \]

This is the adjacent-over-hypotenuse ratio in coordinate form.

Tangent as Vertical Change per Horizontal Change

Since

\[ y=r\sin\theta \qquad\text{and}\qquad x=r\cos\theta, \]

we have

\[ \frac{y}{x} = \frac{r\sin\theta}{r\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta. \]

Thus tangent compares vertical change directly with horizontal change.

Trigonometric Ratios Are Dimensionless

If opposite and hypotenuse are both measured in meters, then

\[ \sin\theta = \frac{6\text{ m}}{10\text{ m}} = 0.6. \]

The units cancel.

Sine, cosine, and tangent are therefore dimensionless ratios.

Scale Does Not Change the Ratios

Compare triangles with side lengths

\[ 3,\ 4,\ 5 \]

and

\[ 6,\ 8,\ 10. \]

The second triangle is twice as large, but the corresponding ratios remain the same:

\[ \frac35=\frac6{10}, \qquad \frac45=\frac8{10}, \qquad \frac34=\frac68. \]

Trigonometric ratios capture shape rather than size.

Using Sine to Find a Side

Suppose

\[ \theta=33^\circ, \qquad H=18. \]

If the opposite side is \(x\), then

\[ \sin33^\circ=\frac{x}{18}. \]

Therefore,

\[ x=18\sin33^\circ\approx9.80. \]

Using Cosine to Find a Side

Suppose

\[ \theta=58^\circ, \qquad H=22. \]

If the adjacent side is \(x\), then

\[ \cos58^\circ=\frac{x}{22}. \]

Thus,

\[ x=22\cos58^\circ\approx11.66. \]

Using Tangent to Find a Side

Suppose an angle is \(24^\circ\) and the adjacent side is \(15\).

If the opposite side is \(x\), then

\[ \tan24^\circ=\frac{x}{15}. \]

Therefore,

\[ x=15\tan24^\circ\approx6.68. \]

Choosing between Sine, Cosine, and Tangent

The correct function depends on the two side roles involved.

The most efficient choice is the ratio that includes the known quantity and the unknown quantity without introducing an unnecessary third side.

Example: Choosing the Efficient Ratio

Suppose a right triangle has an angle of \(40^\circ\), an adjacent side of \(9\), and an unknown opposite side \(x\).

Tangent directly connects the two relevant legs:

\[ \tan40^\circ=\frac{x}{9}. \]

Using sine or cosine would first require finding the hypotenuse, adding an unnecessary step.

Applied Interpretation: Height from Distance

Suppose an observer stands \(60\) meters from a tower and measures an angle of elevation of \(29^\circ\).

Ignoring observer height,

\[ \tan29^\circ=\frac{h}{60}. \]

Therefore,

\[ h=60\tan29^\circ\approx33.26\text{ m}. \]

Tangent is natural because the problem compares vertical height with horizontal distance.

Applied Interpretation: Component of a Force

Suppose a force of magnitude \(F\) acts at an acute angle \(\theta\) above the horizontal.

The horizontal component is

\[ F_x=F\cos\theta, \]

while the vertical component is

\[ F_y=F\sin\theta. \]

This is the same right-triangle geometry used in vector decomposition.

Applied Interpretation: Direction and Motion

If an object moves a distance \(d\) at angle \(\theta\) above the horizontal, its displacement components are

\[ x=d\cos\theta \]

and

\[ y=d\sin\theta. \]

Sine and cosine therefore convert a magnitude and direction into horizontal and vertical quantities.

Comparing Values without a Calculator

For acute angles,

\[ \sin30^\circ<\sin60^\circ \]

because sine increases as the angle increases.

Meanwhile,

\[ \cos30^\circ>\cos60^\circ \]

because cosine decreases as the angle increases.

Understanding behavior can often answer comparison questions without numerical calculation.

Estimating from Nearby Special Angles

Since

\[ \sin30^\circ=\frac12 \]

and sine increases for acute angles,

\[ \sin35^\circ>\frac12. \]

Since \(35^\circ<45^\circ\),

\[ \sin35^\circ<\sin45^\circ=\frac{\sqrt2}{2}. \]

Thus even before using a calculator,

\[ \frac12<\sin35^\circ<\frac{\sqrt2}{2}. \]

Checking a Sine or Cosine Result

If a calculator or algebraic step produces

\[ \sin40^\circ=1.3, \]

the result cannot be correct for an acute angle.

Similarly,

\[ \cos25^\circ=-0.8 \]

cannot be correct in the right-triangle setting because cosine is positive for acute angles.

Checking Tangent with Sine and Cosine

Suppose a calculator gives

\[ \sin37^\circ\approx0.6018 \]

and

\[ \cos37^\circ\approx0.7986. \]

Then

\[ \frac{0.6018}{0.7986}\approx0.7536, \]

which agrees with

\[ \tan37^\circ\approx0.7536. \]

Checking Sine and Cosine with the Pythagorean Identity

If

\[ \sin\theta=\frac35 \qquad\text{and}\qquad \cos\theta=\frac45, \]

then

\[ \left(\frac35\right)^2+\left(\frac45\right)^2 = \frac9{25}+\frac{16}{25} = 1. \]

The identity confirms that the two ratios are consistent.

Common Error: Swapping Sine and Cosine

If the opposite side is \(5\), the adjacent side is \(12\), and the hypotenuse is \(13\), then

\[ \sin\theta=\frac5{13} \]

and

\[ \cos\theta=\frac{12}{13}. \]

Swapping the two ratios changes the angle being described.

Common Error: Using the Hypotenuse in Tangent

Tangent is

\[ \tan\theta=\frac{O}{A}. \]

The hypotenuse does not appear in the tangent ratio.

Writing

\[ \tan\theta=\frac{O}{H} \]

mistakenly uses the sine ratio.

Common Error: Forgetting the Reference Angle

Opposite and adjacent are not permanent names attached to particular sides.

They are determined by the reference angle.

Before using SOH-CAH-TOA, mark the angle and assign the side roles from that angle's perspective.

Common Error: Confusing \(\sin^{-1}\theta\) with \(1/\sin\theta\)

Inverse sine,

\[ \sin^{-1}(x), \]

recovers an angle from a sine value.

The reciprocal of sine is

\[ \frac{1}{\sin\theta}. \]

The reciprocal has a separate trigonometric name, which will be introduced in the next lesson.

Common Error: Squaring the Angle Instead of the Function Value

The expression

\[ \sin^2\theta \]

means

\[ (\sin\theta)^2. \]

It does not mean

\[ \sin(\theta^2). \]

This distinction is essential when using identities.

Common Error: Treating Approximate Values as Exact

A statement such as

\[ \sin37^\circ\approx0.6018 \]

uses the approximation symbol because the decimal has been rounded.

Writing an equals sign would incorrectly suggest that the finite decimal is the exact value.

Common Error: Rounding Intermediate Values Too Aggressively

If a calculation requires

\[ x=\frac{12}{\cos37^\circ}, \]

entering the complete expression is preferable to replacing \(\cos37^\circ\) with a rough value such as \(0.8\) too early.

Preserve precision until the final step whenever practical.

A Reliable Procedure for Evaluating a Trigonometric Function

  1. Identify the angle.
  2. Determine whether the angle is in degrees or radians.
  3. If using a triangle, identify opposite, adjacent, and hypotenuse.
  4. Choose sine, cosine, or tangent.
  5. Use an exact special-angle value when available.
  6. Otherwise evaluate with a calculator in the correct angle mode.
  7. Round only as required.
  8. Check that the result is reasonable for the angle and function.

A Reliable Procedure for Finding a Missing Side

  1. Choose the reference angle.
  2. Label the side roles.
  3. Identify the known side and unknown side.
  4. Choose the ratio containing those two side roles.
  5. Write the trigonometric equation.
  6. Solve algebraically.
  7. Evaluate numerically if needed.
  8. Restore units and check reasonableness.

A Reliable Procedure for Finding an Acute Angle

  1. Choose the unknown angle as the reference angle.
  2. Identify two known side lengths.
  3. Form the corresponding sine, cosine, or tangent ratio.
  4. Apply the matching inverse function.
  5. Use the correct calculator angle mode.
  6. Confirm that the answer lies between \(0^\circ\) and \(90^\circ\).

Practice 1: Evaluate from a Triangle

A right triangle has opposite side \(8\), adjacent side \(15\), and hypotenuse \(17\).

Find \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) for the angle opposite the side of length \(8\).

Practice 2: Find the Remaining Functions

Suppose \(\theta\) is acute and

\[ \sin\theta=\frac{7}{25}. \]

Find \(\cos\theta\) and \(\tan\theta\).

Practice 3: Use the Pythagorean Identity

Suppose \(\theta\) is acute and

\[ \cos\theta=\frac{5}{13}. \]

Find \(\sin\theta\).

Practice 4: Exact Special-Angle Values

Evaluate exactly:

\[ \sin30^\circ,\qquad \cos60^\circ,\qquad \tan45^\circ. \]

Practice 5: Compare without a Calculator

Which is larger?

\[ \sin35^\circ \qquad\text{or}\qquad \sin55^\circ \]

Explain using the behavior of sine on acute angles.

Practice 6: Find a Missing Side

A right triangle has hypotenuse \(20\) and an acute angle of \(43^\circ\).

Find the side adjacent to the \(43^\circ\) angle.

Practice 7: Use Tangent

A right triangle has an acute angle of \(31^\circ\) and adjacent side \(14\).

Find the opposite side.

Practice 8: Recover an Angle

Suppose

\[ \tan\theta=\frac{9}{12}. \]

Find \(\theta\) to the nearest tenth of a degree.

Practice 9: Slope and Tangent

A line rises \(7\) units for every \(10\) units of horizontal run.

Find the tangent of its acute angle of inclination and estimate the angle.

Practice 10: Verify an Identity Numerically

Use

\[ \sin\theta=\frac35 \qquad\text{and}\qquad \cos\theta=\frac45 \]

to verify

\[ \sin^2\theta+\cos^2\theta=1. \]

Practice Answers

Practice 1

\[ \boxed{\sin\theta=\frac8{17}} \] \[ \boxed{\cos\theta=\frac{15}{17}} \] \[ \boxed{\tan\theta=\frac8{15}}. \]

Practice 2

If \(O=7\) and \(H=25\), then

\[ A=\sqrt{25^2-7^2} = \sqrt{576} = 24. \]

Therefore,

\[ \boxed{\cos\theta=\frac{24}{25}} \]

and

\[ \boxed{\tan\theta=\frac7{24}}. \]

Practice 3

\[ \sin^2\theta+\left(\frac5{13}\right)^2=1. \] \[ \sin^2\theta = 1-\frac{25}{169} = \frac{144}{169}. \]

Since \(\theta\) is acute,

\[ \boxed{\sin\theta=\frac{12}{13}}. \]

Practice 4

\[ \boxed{\sin30^\circ=\frac12} \] \[ \boxed{\cos60^\circ=\frac12} \] \[ \boxed{\tan45^\circ=1}. \]

Practice 5

Sine increases as an acute angle increases. Since

\[ 55^\circ>35^\circ, \]

we have

\[ \boxed{\sin55^\circ>\sin35^\circ}. \]

Practice 6

\[ \cos43^\circ=\frac{x}{20}. \] \[ x=20\cos43^\circ\approx\boxed{14.63}. \]

Practice 7

\[ \tan31^\circ=\frac{x}{14}. \] \[ x=14\tan31^\circ\approx\boxed{8.41}. \]

Practice 8

\[ \theta=\tan^{-1}\left(\frac9{12}\right) =\tan^{-1}(0.75). \] \[ \boxed{\theta\approx36.9^\circ}. \]

Practice 9

\[ \tan\theta=\frac7{10}=0.7. \] \[ \theta=\tan^{-1}(0.7)\approx\boxed{35.0^\circ}. \]

Practice 10

\[ \sin^2\theta+\cos^2\theta = \left(\frac35\right)^2+\left(\frac45\right)^2. \] \[ = \frac9{25}+\frac{16}{25} = \frac{25}{25} = \boxed{1}. \]

What to Remember

The three primary trigonometric functions begin with

\[ \boxed{ \sin\theta=\frac{O}{H}, \qquad \cos\theta=\frac{A}{H}, \qquad \tan\theta=\frac{O}{A} }. \]

Where We Go Next

Sine, cosine, and tangent form the core of elementary trigonometry, but they are only half of the classical six trigonometric functions.

The next lesson, Reciprocal Trigonometric Relationships, introduces cosecant, secant, and cotangent and shows how all six functions are linked through reciprocal structure.