Finding the Opposite Side (Using Sine)

A ladder leans against a wall at an angle of $65°$ with the ground. If the ladder is $4$ meters long, how high up the wall does it reach?

Draw and label the triangle: The ladder is the hypotenuse ($4$ m), and we want the height (opposite to the $65°$ angle) = Known: hypotenuse = $4$ m, angle = $65°$

Choose the right ratio: We have hypotenuse and want opposite, so use $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ = Use sine

Set up the equation: $\sin(65°) = \frac{h}{4}$ = Equation ready

Solve for the unknown: $h = 4 \times \sin(65°) = 4 \times 0.906 = 3.624$ = $h \approx 3.62$ m

Answer: The ladder reaches approximately $3.62$ meters up the wall.

Finding the Adjacent Side (Using Cosine)

A ship sails $12$ km on a bearing that makes a $40°$ angle with the north-south line. How far north has the ship traveled?

Identify the triangle parts: Distance sailed = hypotenuse ($12$ km), northward distance = adjacent to the $40°$ angle = Known: hypotenuse = $12$ km, angle = $40°$

Choose the right ratio: We have hypotenuse and want adjacent, so use $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ = Use cosine

Set up the equation: $\cos(40°) = \frac{d}{12}$ = Equation ready

Solve for the unknown: $d = 12 \times \cos(40°) = 12 \times 0.766 = 9.192$ = $d \approx 9.19$ km

Answer: The ship has traveled approximately $9.19$ km north.

Finding a Side Using Tangent

From a point $50$ meters away from the base of a building, the angle of elevation to the top is $32°$. How tall is the building?

Identify the triangle parts: Distance from building = adjacent ($50$ m), building height = opposite to the $32°$ angle = Known: adjacent = $50$ m, angle = $32°$

Choose the right ratio: We have adjacent and want opposite, so use $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ = Use tangent

Set up the equation: $\tan(32°) = \frac{h}{50}$ = Equation ready

Solve for the unknown: $h = 50 \times \tan(32°) = 50 \times 0.625 = 31.25$ = $h \approx 31.25$ m

Answer: The building is approximately $31.25$ meters tall.

Finding the Hypotenuse

A wheelchair ramp rises $0.8$ meters over a horizontal distance. If the ramp makes a $5°$ angle with the ground, how long is the ramp?

Identify the triangle parts: Rise = opposite ($0.8$ m), ramp length = hypotenuse (unknown), angle = $5°$ = Known: opposite = $0.8$ m, angle = $5°$

Choose the right ratio: We have opposite and want hypotenuse, so use $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ = Use sine

Set up the equation: $\sin(5°) = \frac{0.8}{L}$ = Equation ready

Solve for the unknown: $L = \frac{0.8}{\sin(5°)} = \frac{0.8}{0.0872} = 9.17$ = $L \approx 9.17$ m

Answer: The ramp is approximately $9.17$ meters long.

Mistake: Using the wrong trigonometric ratio

Why: Students often mix up which ratio to use when they have different combinations of sides.

Correct: Always label sides first (opposite, adjacent, hypotenuse), then choose the ratio that contains both your known and unknown sides.

Mistake: Confusing which side is opposite vs adjacent

Why: Opposite and adjacent depend on which angle you're using, not fixed positions in the triangle.

Correct: The opposite side is across from your angle. The adjacent side is next to your angle (but not the hypotenuse).

Mistake: Calculator in wrong mode (radians instead of degrees)

Why: Calculators can work in degrees or radians, and using the wrong mode gives completely wrong answers.

Correct: Make sure your calculator is set to DEG mode when working with degree angles.

Mistake: Dividing instead of multiplying (or vice versa)

Why: When solving $\sin(\theta) = \frac{x}{h}$, students sometimes set up the algebra incorrectly.

Correct: To find the numerator, multiply: $x = h \times \sin(\theta)$. To find the denominator, divide: $h = \frac{x}{\sin(\theta)}$.

Architecture and Construction

Architects and builders use trigonometry to calculate roof pitches, ramp lengths, and support beam dimensions.

A roof must have a $30°$ pitch and span $8$ meters horizontally. Using cosine, the rafter length is $\frac{8}{\cos(30°)} \approx 9.24$ meters.

Navigation and Aviation

Pilots and navigators calculate distances and altitudes using trigonometry when they know angles and partial distances.

A plane descends at a $3°$ glide angle from $10{,}000$ feet altitude. The horizontal distance to the runway is $\frac{10000}{\tan(3°)} \approx 190{,}810$ feet (about 36 miles).

Q: How do I know which trig ratio to use?

A: First label your sides relative to the angle: opposite (across from angle), adjacent (next to angle, not hypotenuse), and hypotenuse (longest side). Then pick the ratio that contains your known side and the side you want to find.

Q: What if I get a very large or very small answer?

A: Check your calculator mode! If it's in radians instead of degrees, your answers will be wrong. Also verify you're using the correct ratio and doing the algebra correctly.

Q: Can I use any angle in the triangle?

A: You should use one of the acute angles (not the 90-degree angle). Either acute angle will work, but your 'opposite' and 'adjacent' sides will switch depending on which angle you choose.