Teacher Guide: Finding Missing Sides
Learn how to use trigonometric ratios to calculate unknown side lengths in right triangles.
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Class quiz
10 questions on Trig Applications. Students join with a name, you see everyone's score.
For Teachers
- Identify opposite, adjacent, and hypotenuse sides relative to a given angle
- Select the appropriate trigonometric ratio for a given problem
- Calculate missing side lengths using sine, cosine, and tangent
- Solve real-world problems involving right triangles and unknown sides
- • Understanding of right triangles and the Pythagorean theorem
- • Knowledge of sine, cosine, and tangent ratios (SOH CAH TOA)
- • Basic algebraic equation solving
- • Ability to use a scientific calculator for trig functions
- 1. Why do we need trigonometry when we already have the Pythagorean theorem?
- 2. What careers use finding missing sides with trigonometry regularly?
- 3. How could you measure the height of a tree without climbing it?
- 4. Why is it important that calculators be in the correct mode (degrees vs radians)?
The opposite and adjacent sides are always in the same position
You can only use trigonometry to find the hypotenuse
Larger angles always mean larger sides
For Struggling Students:
- • Provide a decision flowchart for choosing the correct trig ratio
- • Use color-coding to identify opposite (red), adjacent (blue), hypotenuse (green)
- • Start with integer answers before moving to decimal calculations
For On-Level Students:
- • Solve problems with various angle positions
- • Include word problems requiring diagram drawing
- • Practice problems where the unknown is in the denominator
For Advanced Students:
- • Combine with Pythagorean theorem for multi-step problems
- • Introduce problems with angles of elevation and depression
- • Solve for sides when given two angles (using angle sum property)
- HSG-SRT.C.8 (CCSS.MATH.CONTENT.HSG.SRT.C.8)
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
- HSG-SRT.C.6 (CCSS.MATH.CONTENT.HSG.SRT.C.6)
Understand that by similarity, side ratios in right triangles are properties of the angles
- visualInteractive Triangle Solver
Students adjust angles and sides to see how trig ratios work
- activityMeasure Your School
Use clinometers to measure building heights without climbing
- worksheetReal-World Trig Problems
Practice problems featuring ladders, ramps, and shadows
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
Worked Examples
A ladder leans against a wall at an angle of with the ground. If the ladder is meters long, how high up the wall does it reach?
Draw and label the triangle
The ladder is the hypotenuse ( m), and we want the height (opposite to the angle) → Known: hypotenuse = m, angle =
Choose the right ratio
We have hypotenuse and want opposite, so use → Use sine
Set up the equation
→ Equation ready
Solve for the unknown
→ m
Answer: The ladder reaches approximately meters up the wall.
Common Mistakes
Using the wrong trigonometric ratio
Why it's wrong: 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.
Confusing which side is opposite vs adjacent
Why it's wrong: 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).
Calculator in wrong mode (radians instead of degrees)
Why it's wrong: 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.
Dividing instead of multiplying (or vice versa)
Why it's wrong: When solving , students sometimes set up the algebra incorrectly.
Correct: To find the numerator, multiply: . To find the denominator, divide: .
Why It Matters
- Construction: Calculating rafter lengths for roofs at specific angles
- Navigation: Determining distances when only angles are known
- Surveying: Measuring heights of buildings or mountains from a distance
- Engineering: Designing ramps, bridges, and structures with precise measurements
- Aviation: Calculating glide paths and approach angles
Real World Applications
Architecture and Construction
Architects and builders use trigonometry to calculate roof pitches, ramp lengths, and support beam dimensions.
Example:
A roof must have a pitch and span meters horizontally. Using cosine, the rafter length is meters.
You need to build a roof with a pitch. The horizontal span is meters.
How long should each rafter be?
Step 1: Write the mathematical expression
Use cosine: rafter = horizontal / cos(angle)
Navigation and Aviation
Pilots and navigators calculate distances and altitudes using trigonometry when they know angles and partial distances.
Example:
A plane descends at a glide angle from feet altitude. The horizontal distance to the runway is feet (about 36 miles).
A helicopter needs to reach a rooftop meters away horizontally. The angle of elevation from the helipad is .
How high is the rooftop above the helipad?
Step 1: Write the mathematical expression
Use tangent: height = horizontal distance times tan(angle)
Key Takeaways
- 1To find a missing side, identify which sides you have (opposite, adjacent, hypotenuse) relative to the known angle
- 2Use SOH CAH TOA to select the correct ratio for your known and unknown sides
- 3To find the numerator of a ratio, multiply:
- 4To find the denominator, divide:
- 5Always check that your calculator is in degree mode when using degree measurements
Frequently Asked Questions
How do I know which trig ratio to use?
What if I get a very large or very small answer?
Can I use any angle in the triangle?
Glossary
- Opposite side
- The side of a right triangle that is across from (not touching) the reference angle
- Adjacent side
- The side of a right triangle that is next to the reference angle (not the hypotenuse)
- Hypotenuse
- The longest side of a right triangle, opposite the right angle
- Angle of elevation
- The angle measured upward from the horizontal to a line of sight
- Angle of depression
- The angle measured downward from the horizontal to a line of sight
Formula Card
Opposite from Hypotenuse
When you know the hypotenuse and need the opposite side
Adjacent from Hypotenuse
When you know the hypotenuse and need the adjacent side
Opposite from Adjacent
When you know the adjacent side and need the opposite side
Hypotenuse from Opposite
When you know the opposite side and need the hypotenuse
Hypotenuse from Adjacent
When you know the adjacent side and need the hypotenuse
Adjacent from Opposite
When you know the opposite side and need the adjacent side