The Central Idea

A right triangle contains one \(90^\circ\) angle. The other two angles are acute, and their measures add to \(90^\circ\).

If we focus on one acute angle, say \(\theta\), the side lengths form three important ratios:

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

These ratios depend on the angle, not on the overall size of the triangle. That fact is the foundation of right-triangle trigonometry.

Why the Ratios Stay the Same

Imagine two right triangles that both contain the same acute angle \(\theta\). Because each triangle also contains a right angle, the third angle must be the same in both triangles.

The triangles therefore have the same shape and are similar.

Corresponding side lengths may be larger or smaller, but they scale by the same factor. Ratios of corresponding sides therefore remain unchanged.

This is why a \(30^\circ\) angle has the same sine, cosine, and tangent in every right triangle containing a \(30^\circ\) angle.

The Hypotenuse

The hypotenuse is the side opposite the right angle.

It is always the longest side of a right triangle.

The hypotenuse does not depend on which acute angle we are studying. Its identity is fixed by the location of the \(90^\circ\) angle.

The Opposite and Adjacent Sides

The words opposite and adjacent are relative to the acute angle we choose.

For an acute angle \(\theta\):

If we switch to the other acute angle, the opposite and adjacent sides switch roles.

Side Roles Depend on the Reference Angle

Suppose a right triangle has acute angles \(\theta\) and \(\phi\).

A leg that is opposite \(\theta\) is adjacent to \(\phi\), while a leg that is adjacent to \(\theta\) is opposite \(\phi\).

The hypotenuse remains the hypotenuse for both angles.

Before writing any trigonometric ratio, identify the reference angle first.

Sine

The sine of an acute angle compares the opposite side with the hypotenuse:

\[ \boxed{\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}}. \]

If the side opposite \(\theta\) has length \(6\) and the hypotenuse has length \(10\), then

\[ \sin\theta=\frac{6}{10}=\frac{3}{5}=0.6. \]

Cosine

The cosine of an acute angle compares the adjacent side with the hypotenuse:

\[ \boxed{\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}}. \]

If the adjacent side has length \(8\) and the hypotenuse has length \(10\), then

\[ \cos\theta=\frac{8}{10}=\frac{4}{5}=0.8. \]

Tangent

The tangent of an acute angle compares the opposite and adjacent legs:

\[ \boxed{\tan\theta=\frac{\text{opposite}}{\text{adjacent}}}. \]

If the opposite side has length \(6\) and the adjacent side has length \(8\), then

\[ \tan\theta=\frac{6}{8}=\frac{3}{4}=0.75. \]

SOH-CAH-TOA

A common memory aid is SOH-CAH-TOA:

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

Here \(O\) means opposite, \(A\) means adjacent, and \(H\) means hypotenuse.

The mnemonic is useful, but it only works after the side roles have been identified correctly.

The Three Ratios in One Triangle

Consider a right triangle whose side lengths are

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

Let \(\theta\) be the acute angle opposite the side of length \(6\). Then

\[ \sin\theta=\frac{6}{10}=\frac{3}{5}, \] \[ \cos\theta=\frac{8}{10}=\frac{4}{5}, \]

and

\[ \tan\theta=\frac{6}{8}=\frac{3}{4}. \]

The same triangle supplies all three ratios.

Tangent as a Ratio of Sine and Cosine

In a right triangle,

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

Divide sine by cosine:

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

But \(O/A=\tan\theta\). Therefore,

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

Later lessons will treat this as one of the basic trigonometric identities.

The Pythagorean Connection

If the legs of a right triangle are \(a\) and \(b\), and the hypotenuse is \(c\), then

\[ a^2+b^2=c^2. \]

Trigonometry does not replace the Pythagorean relationship. The two tools work together.

The Pythagorean relationship connects the three side lengths, while trigonometric ratios connect side lengths to acute angles.

Choosing the Right Ratio

You do not need to use all three trigonometric ratios in every problem.

Identify:

  1. the reference angle;
  2. the side you know;
  3. the side you want;
  4. the ratio that contains exactly those side roles.

For example, if the known side is adjacent and the unknown side is the hypotenuse, cosine is the natural choice.

Solving for an Unknown Opposite Side

Suppose

\[ \theta=35^\circ, \qquad H=12, \]

and we want the opposite side \(x\).

Use sine:

\[ \sin 35^\circ=\frac{x}{12}. \]

Multiply by \(12\):

\[ x=12\sin35^\circ. \]

Numerically,

\[ x\approx 6.88. \]

The exact calculator value depends on the desired rounding.

Solving for an Unknown Adjacent Side

Suppose

\[ \theta=42^\circ, \qquad H=15, \]

and the adjacent side is \(x\).

Use cosine:

\[ \cos42^\circ=\frac{x}{15}. \]

Therefore,

\[ x=15\cos42^\circ\approx 11.15. \]

Solving for an Unknown Leg with Tangent

Suppose an angle is \(28^\circ\), the adjacent leg is \(9\), and the opposite leg is \(x\).

Tangent relates the two legs:

\[ \tan28^\circ=\frac{x}{9}. \]

Thus,

\[ x=9\tan28^\circ\approx 4.79. \]

Solving for the Hypotenuse

Suppose

\[ \theta=50^\circ, \qquad O=7, \]

and the hypotenuse is \(h\).

Use sine:

\[ \sin50^\circ=\frac{7}{h}. \]

Multiply by \(h\):

\[ h\sin50^\circ=7. \]

Divide by \(\sin50^\circ\):

\[ \boxed{h=\frac{7}{\sin50^\circ}\approx 9.14}. \]

Algebra Still Matters

Trigonometric problems often reduce to ordinary algebra after the correct ratio is written.

For example,

\[ \cos\theta=\frac{a}{x} \]

can be solved for \(x\):

\[ x\cos\theta=a, \] \[ \boxed{x=\frac{a}{\cos\theta}}. \]

A common source of errors is not the trigonometry itself but rearranging the resulting equation incorrectly.

Finding an Angle from Side Lengths

Trigonometric ratios can also be used in reverse.

If

\[ \sin\theta=\frac{3}{5}, \]

then the angle can be found with the inverse sine:

\[ \theta=\sin^{-1}\left(\frac{3}{5}\right). \]

In degree mode,

\[ \theta\approx 36.87^\circ. \]

Inverse Sine, Cosine, and Tangent

To recover an acute angle from a side ratio, use an inverse trigonometric function:

\[ \boxed{ \begin{aligned} \theta&=\sin^{-1}\left(\frac{O}{H}\right),\\ \theta&=\cos^{-1}\left(\frac{A}{H}\right),\\ \theta&=\tan^{-1}\left(\frac{O}{A}\right). \end{aligned} } \]

The notation \(\sin^{-1}\) means inverse sine here. It does not mean the reciprocal \(1/\sin\theta\).

Reciprocal trigonometric relationships will be treated separately in a later lesson.

Example: Finding an Angle with Tangent

Suppose the opposite leg is \(5\) and the adjacent leg is \(12\).

Then

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

Apply inverse tangent:

\[ \theta=\tan^{-1}\left(\frac{5}{12}\right). \]

In degree mode,

\[ \boxed{\theta\approx22.62^\circ}. \]

The Two Acute Angles Are Complementary

In every right triangle,

\[ \theta+\phi+90^\circ=180^\circ. \]

Therefore,

\[ \boxed{\theta+\phi=90^\circ}. \]

Once one acute angle is known, the other can be found by subtraction:

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

Complementary-Angle Relationships

Because opposite and adjacent sides switch roles when we move from one acute angle to the other,

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

and

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

These relationships are called cofunction relationships. They will become even more useful when trigonometric functions are studied beyond right triangles.

Special Triangle: \(45^\circ\)-\(45^\circ\)-\(90^\circ\)

A \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle is isosceles, so its two legs have equal length.

If each leg has length \(1\), the Pythagorean relationship gives

\[ 1^2+1^2=h^2, \] \[ h=\sqrt2. \]

Therefore,

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

and

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

Special Triangle: \(30^\circ\)-\(60^\circ\)-\(90^\circ\)

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

\[ \boxed{1:\sqrt3:2}. \]

The side opposite \(30^\circ\) is the shortest, the side opposite \(60^\circ\) has length \(\sqrt3\) times the short leg, and the hypotenuse is twice the short leg.

Therefore,

\[ \sin30^\circ=\frac12, \qquad \cos30^\circ=\frac{\sqrt3}{2}, \qquad \tan30^\circ=\frac{1}{\sqrt3}=\frac{\sqrt3}{3}. \]

For \(60^\circ\),

\[ \sin60^\circ=\frac{\sqrt3}{2}, \qquad \cos60^\circ=\frac12, \qquad \tan60^\circ=\sqrt3. \]

Exact Values and Decimal Values

Some trigonometric values are known exactly. For example,

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

is exact.

A calculator may display

\[ 0.5, \]

which represents the same value.

Other angles usually produce decimal approximations. Unless a problem asks for a specific rounding, it is often best to keep the trigonometric expression exact until the final step.

Degree Mode and Radian Mode

Calculator mode matters.

If an angle is written as \(35^\circ\), the calculator should be in degree mode.

If an angle is written in radians, the calculator should be in radian mode.

For example,

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

but entering \(30\) while the calculator is in radian mode evaluates \(\sin(30\text{ radians})\), which is a completely different input.

Rounding Strategy

Avoid unnecessary rounding in the middle of a calculation.

Suppose

\[ x=\frac{17}{\cos38^\circ}. \]

It is generally better to enter the full expression into a calculator and round once at the end rather than first rounding \(\cos38^\circ\).

Early rounding can accumulate error.

Angle of Elevation

An angle of elevation is measured upward from a horizontal line of sight.

Suppose a person stands \(40\) meters from the base of a building and measures an angle of elevation of \(32^\circ\) to the roof.

Ignoring eye height for the moment, the building height \(h\) satisfies

\[ \tan32^\circ=\frac{h}{40}. \]

Therefore,

\[ h=40\tan32^\circ\approx25.0\text{ m}. \]

Including Observer Height

In a real measurement problem, the measuring instrument may be above the ground.

If the instrument is \(1.6\) meters high and the vertical rise from the instrument to the top of the building is \(25.0\) meters, then the total height is

\[ 25.0+1.6=26.6\text{ m}. \]

A diagram should distinguish the vertical change measured by the triangle from the complete physical height.

Angle of Depression

An angle of depression is measured downward from a horizontal line of sight.

If an observer looks downward from a cliff toward an object below, the angle of depression and the corresponding angle of elevation are equal when the horizontal lines are parallel.

This often allows the same right-triangle ratios to be used without introducing a new formula.

Indirect Measurement

Trigonometry is especially useful when a distance cannot be measured directly.

Examples include:

The central strategy is to model the situation with a right triangle and identify which measured quantities correspond to the triangle's sides and angles.

Example: Ladder Against a Wall

A \(20\)-foot ladder makes a \(68^\circ\) angle with level ground.

If \(h\) is the vertical height reached on the wall, then \(h\) is opposite the \(68^\circ\) angle and \(20\) is the hypotenuse.

\[ \sin68^\circ=\frac{h}{20}. \]

Therefore,

\[ \boxed{h=20\sin68^\circ\approx18.54\text{ ft}}. \]

Example: Horizontal Distance

Suppose a \(15\)-meter cable runs from the ground to the top of a pole and makes a \(55^\circ\) angle with the ground.

Let \(d\) be the horizontal distance from the anchor point to the pole.

Here \(d\) is adjacent to \(55^\circ\), so

\[ \cos55^\circ=\frac{d}{15}. \]

Thus,

\[ \boxed{d=15\cos55^\circ\approx8.60\text{ m}}. \]

Example: Ramp Angle

A ramp rises \(2.5\) feet over a horizontal run of \(18\) feet.

If \(\theta\) is the angle the ramp makes with the horizontal, then

\[ \tan\theta=\frac{2.5}{18}. \]

Therefore,

\[ \boxed{\theta=\tan^{-1}\left(\frac{2.5}{18}\right)\approx7.91^\circ}. \]

Slope and Tangent

In coordinate geometry, slope is

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

If a line makes an acute angle \(\theta\) with the positive horizontal direction, then

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

Therefore,

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

This is an early connection between trigonometry and analytic geometry.

Using More Than One Method

Some problems can be solved in more than one way.

If a right triangle has hypotenuse \(13\) and one leg \(5\), the other leg can be found with the Pythagorean relationship:

\[ 5^2+x^2=13^2. \]

This gives

\[ x=12. \]

Afterward, trigonometry can find either acute angle.

Efficient problem solving means choosing the simplest valid tool, not forcing trigonometry into every step.

Solving a Right Triangle

To solve a triangle means to determine all unknown side lengths and angle measures.

A right triangle already has one known angle:

\[ 90^\circ. \]

Depending on the information given, the remaining quantities may be found with:

Example: Solve a Right Triangle

Suppose a right triangle has hypotenuse \(10\) and one leg \(6\).

First find the other leg:

\[ 6^2+b^2=10^2, \] \[ b^2=64, \] \[ b=8. \]

Let \(\theta\) be opposite the side of length \(6\). Then

\[ \sin\theta=\frac{6}{10}=\frac35. \]

Therefore,

\[ \theta\approx36.87^\circ. \]

The other acute angle is

\[ 90^\circ-36.87^\circ\approx53.13^\circ. \]

Reasonableness Checks

A result should make geometric sense.

These checks can catch mistakes before a final answer is accepted.

Checking with the Pythagorean Relationship

Suppose trigonometry gives approximate leg lengths

\[ a\approx7.66, \qquad b\approx6.43 \]

for a triangle with hypotenuse \(10\).

A consistency check is

\[ 7.66^2+6.43^2\approx100. \]

Small differences may appear because the side lengths were rounded.

Checking with Another Trigonometric Ratio

If a calculation gives

\[ O=6,\qquad A=8,\qquad H=10, \]

then for the angle opposite the side of length \(6\),

\[ \sin\theta=\frac{3}{5}, \qquad \cos\theta=\frac{4}{5}. \]

These values should correspond to the same angle. Comparing independent relationships is a useful verification strategy.

Common Error: Misidentifying the Hypotenuse

The hypotenuse is not simply the side that looks longest in a sketch.

It is defined as the side opposite the \(90^\circ\) angle.

Diagrams are not always drawn to scale, so use the angle marking rather than visual appearance.

Common Error: Treating Opposite and Adjacent as Fixed Labels

Opposite and adjacent depend on the reference angle.

A side can be opposite one acute angle and adjacent to the other.

Always mark the angle being used before assigning \(O\) and \(A\).

Common Error: Using the Wrong Ratio

Suppose the known side is adjacent and the unknown side is the hypotenuse.

The relevant ratio is

\[ \cos\theta=\frac{A}{H}, \]

not sine or tangent.

A quick side-role inventory prevents many errors.

Common Error: Inverting a Ratio

If

\[ \sin\theta=\frac{O}{H}, \]

do not reverse the order and write \(H/O\).

In a right triangle with an acute angle, \(O/H<1\), which is consistent with the fact that the opposite leg is shorter than the hypotenuse.

A value greater than \(1\) for the sine or cosine of an acute angle is a strong warning that the ratio has been written incorrectly.

Common Error: Confusing Inverse with Reciprocal

The expression

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

denotes inverse sine in this context.

The reciprocal of sine is

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

These are different operations and should not be confused.

Common Error: Calculator in the Wrong Angle Mode

If the problem uses degrees but the calculator is set to radians, even a correctly written equation can produce an incorrect numerical answer.

Check angle mode before evaluating trigonometric functions or their inverses.

Common Error: Rounding Too Early

Suppose a multi-step problem first gives an angle of

\[ 36.869897\ldots^\circ. \]

Replacing it immediately with \(37^\circ\) and using the rounded value in later calculations can noticeably alter the final result.

Keep full calculator precision through intermediate steps whenever practical.

Common Error: Forgetting Units

Trigonometric ratios themselves are ratios of lengths and therefore have no length unit.

But a solved side length should carry the unit from the problem:

\[ 18.5\text{ ft}, \qquad 26.6\text{ m}, \qquad 4.2\text{ km}. \]

Angle answers should also indicate whether the measure is in degrees or radians when context does not make that clear.

A Reliable Procedure for Finding a Missing Side

  1. Locate the right angle and identify the hypotenuse.
  2. Choose the acute reference angle.
  3. Label the relevant sides as opposite, adjacent, or hypotenuse.
  4. Identify the known side and the unknown side.
  5. Choose sine, cosine, or tangent to connect those two sides.
  6. Substitute the known values.
  7. Solve the resulting algebraic equation.
  8. Round only as instructed or at the final step.
  9. Check that the result is geometrically reasonable.

A Reliable Procedure for Finding an Angle

  1. Choose the unknown acute angle as the reference angle.
  2. Identify the two known side roles.
  3. Form the corresponding sine, cosine, or tangent ratio.
  4. Apply the matching inverse trigonometric function.
  5. Confirm that the calculator is in the intended angle mode.
  6. Round as required.
  7. Check that the result lies between \(0^\circ\) and \(90^\circ\) for an acute angle.

A Reliable Procedure for Applied Problems

  1. Draw or interpret a diagram.
  2. Identify the right triangle inside the situation.
  3. Mark measured quantities and unknown quantities.
  4. Distinguish physical dimensions from the triangle's modeled dimensions.
  5. Choose an appropriate trigonometric relationship.
  6. Solve algebraically.
  7. Restore units.
  8. Account for offsets such as eye height when necessary.
  9. Check whether the answer makes physical sense.

Practice 1: Identify the Side Roles

In a right triangle, angle \(\theta\) is one acute angle. The side opposite the right angle has length \(13\). The side directly across from \(\theta\) has length \(5\).

Identify the hypotenuse, opposite side, and adjacent side. Then find the missing side length.

Practice 2: Find a Missing Opposite Side

A right triangle has

\[ \theta=38^\circ \]

and hypotenuse \(14\). Find the side opposite \(\theta\).

Practice 3: Find a Missing Adjacent Side

A right triangle has

\[ \theta=27^\circ \]

and hypotenuse \(18\). Find the side adjacent to \(\theta\).

Practice 4: Use Tangent

A right triangle has an acute angle of \(41^\circ\). The adjacent leg is \(11\). Find the opposite leg.

Practice 5: Find the Hypotenuse

The side opposite a \(52^\circ\) angle has length \(9\). Find the hypotenuse.

Practice 6: Find an Angle

A right triangle has opposite leg \(7\) and adjacent leg \(24\). Find the acute angle between the adjacent leg and the hypotenuse.

Practice 7: Solve a Right Triangle

A right triangle has legs \(9\) and \(12\).

Find the hypotenuse and both acute angles.

Practice 8: Angle of Elevation

From a point \(50\) meters from the base of a tower, the angle of elevation to the top is \(36^\circ\). Ignore observer height.

Estimate the tower's height.

Practice 9: Ladder

A \(24\)-foot ladder makes a \(72^\circ\) angle with level ground.

How high does the ladder reach on a vertical wall?

Practice 10: Identify the Error

A student is given a right triangle with a \(30^\circ\) angle, opposite side \(4\), and hypotenuse \(8\). The student writes

\[ \cos30^\circ=\frac{4}{8}. \]

Explain the error and write the correct trigonometric relationship.

Practice Answers

Practice 1

The hypotenuse is \(13\), the opposite side is \(5\), and the adjacent side is found from

\[ 5^2+a^2=13^2. \]

Thus,

\[ a^2=144, \qquad \boxed{a=12}. \]

Practice 2

\[ \sin38^\circ=\frac{x}{14}. \] \[ x=14\sin38^\circ\approx\boxed{8.62}. \]

Practice 3

\[ \cos27^\circ=\frac{x}{18}. \] \[ x=18\cos27^\circ\approx\boxed{16.04}. \]

Practice 4

\[ \tan41^\circ=\frac{x}{11}. \] \[ x=11\tan41^\circ\approx\boxed{9.56}. \]

Practice 5

\[ \sin52^\circ=\frac{9}{h}. \] \[ h=\frac{9}{\sin52^\circ}\approx\boxed{11.42}. \]

Practice 6

\[ \tan\theta=\frac{7}{24}. \] \[ \theta=\tan^{-1}\left(\frac{7}{24}\right)\approx\boxed{16.26^\circ}. \]

Practice 7

First,

\[ h=\sqrt{9^2+12^2}=\sqrt{225}=15. \]

Let \(\theta\) be opposite the side of length \(9\):

\[ \sin\theta=\frac{9}{15}=\frac35. \] \[ \theta\approx36.87^\circ. \]

The other acute angle is

\[ 90^\circ-36.87^\circ\approx53.13^\circ. \]

Therefore,

\[ \boxed{h=15,\qquad \theta\approx36.87^\circ,\qquad \phi\approx53.13^\circ}. \]

Practice 8

\[ \tan36^\circ=\frac{h}{50}. \] \[ h=50\tan36^\circ\approx\boxed{36.33\text{ m}}. \]

Practice 9

\[ \sin72^\circ=\frac{h}{24}. \] \[ h=24\sin72^\circ\approx\boxed{22.83\text{ ft}}. \]

Practice 10

The ratio \(4/8\) is opposite over hypotenuse, so it corresponds to sine, not cosine:

\[ \boxed{\sin30^\circ=\frac{4}{8}=\frac12}. \]

Cosine would compare the adjacent side with the hypotenuse.

What to Remember

For a chosen acute angle \(\theta\) in a right triangle:

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

Where We Go Next

Right-triangle trigonometry introduced sine, cosine, and tangent as ratios of side lengths in similar right triangles.

The next lesson, Sine, Cosine, and Tangent, develops these three functions more directly, compares their behavior, and prepares them to move beyond the geometry of a single right triangle.