The Trig Ratio Builder
Sections 7.2, 7.6, 7.7 -- Angles, triangles, Pythagorean Theorem, six trig ratios, solving for missing sides
Step 1 of 9
A ladder leans against a wall. Adjust the angle and watch the three lengths: the ladder (hypotenuse), the height it reaches (opposite), and the distance from the wall (adjacent).
Step 2 of 9
Check the Pythagorean Theorem. Does \( \text{height}^2 + \text{distance}^2 = \text{ladder}^2 \) at every angle? Try at least three different angles.
Step 3 of 9
As the angle steepens (gets larger), predict what happens to each measurement.
Height (opposite side):
Distance from wall (adjacent side):
Ladder length (hypotenuse):
Step 4 of 9
Watch the ratio of height to ladder length. Set the angle to 30 degrees and try different ladder lengths. What do you notice about the ratio?
Try at least 3 different ladder lengths at 30 degrees.
Step 5 of 9
Change the length all you want. At 30 degrees, that ratio is always 0.5. The ratio does not care about size. It only cares about the angle. Now test two more angles to confirm that the ratio depends only on the angle, not the ladder length.
Test at least 2 different angles, each with at least 2 different ladder lengths.
Step 6 of 9
That ratio -- opposite over hypotenuse -- is called the sine of the angle. Now predict: what ratio do you think "adjacent over hypotenuse" describes?
As the angle increases, what happens to the sine (opp/hyp)?
As the angle increases, what happens to the cosine (adj/hyp)?
Step 7 of 9
Two ladders at 30 degrees, different lengths, drawn overlapping. Same shape, different size. That is why the ratio holds. These triangles are similar.
Both triangles have the same angles, so they are similar. Similar triangles have proportional sides. That is why the ratio of opposite to hypotenuse is the same regardless of size.
Step 8 of 9
The three reciprocal ratios are just flips of what you already know. But the names trip everyone up, because the pairings are not alphabetical.
The mismatch that trips everyone up:
cosecant pairs with sine (not cosine).
secant pairs with cosine (not sine).
cotangent pairs with tangent.
The "co-" prefix does NOT mean they go together. Cosecant is the reciprocal of sine. Secant is the reciprocal of cosine.
Which is the reciprocal of cosine?
Step 9 of 9
A 20-foot ladder at 65 degrees. How high does it reach? Set up the equation using sine, then solve for the height.
Which equation gives the height?
Explore the Right Triangle
Adjust the angle and hypotenuse. All six trig ratios update live.
SOHCAHTOA
\( \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \) \( \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \) \( \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
Adjust the angle or hypotenuse to see how the six ratios change.
Try This
You are stringing lights from a roof hook 12 feet up to a pole 25 feet away across the yard. What angle does the string make with the ground? How long a string do you need? Buy the next whole foot up.
Enter the angle within 0.5 degrees, and the string length within 0.5 feet.
An ADA-compliant ramp can rise at most 1 foot for every 12 feet of run. A contractor built one rising 3 feet over 36 feet of run. What is the angle? Does it comply? What is the maximum allowed angle?
Enter angles within 0.1 degrees.
Does this ramp comply with ADA?
A cell tower casts a 95-foot shadow when the sun sits at 52 degrees above the horizon. How tall is the tower? Two hours later the shadow is 140 feet. What is the sun's elevation angle now? Did the tower change?
Enter the tower height within 1 foot, the angle within 0.5 degrees.
Did the tower change height?
- Verify the Pythagorean Theorem for right triangles
- Discover that trig ratios depend on the angle, not on the triangle's size
- Connect the six trig ratios to the sides of a right triangle
- Understand why similar triangles make trig ratios work
- Identify the reciprocal pairings (sin/csc, cos/sec, tan/cot)
- Solve for a missing side using a trig ratio
Quick Check
A 15-foot ladder leans against a wall at 50 degrees. Which expression gives the height it reaches on the wall?
Instructor Notes
Teaching Notes
This simulation delays naming the trig ratios until after students discover that the ratio of opposite to hypotenuse depends only on the angle. The name "sine" arrives in Step 6, after students have already seen the invariance in Steps 4 and 5. This is deliberate: the concept precedes the vocabulary.
The ladder context keeps the triangle grounded in a physical object. Students can reason about "the height the ladder reaches" more naturally than about "the opposite side."
Common Student Errors
- Believing the trig ratio depends on triangle size. Steps 4 and 5 address this directly by having students change the ladder length while watching the ratio hold steady.
- Pairing secant with sine and cosecant with cosine. Step 8 names this mismatch explicitly. The "co-" prefix is misleading, and students need to be told directly that cosecant goes with sine.
- Using degrees in calculator functions that expect radians. All computation in this simulation uses degree-to-radian conversion internally. Point this out if students get unexpected results on their own calculators.
Discussion Questions
- Why does the ratio stay the same when you change the ladder length but keep the angle fixed?
- If you know the sine of an angle, can you figure out the cosine without a calculator? How?
- A 30-60-90 triangle and a 45-45-90 triangle are called "special." What makes their ratios special enough to memorize?
Exam Connection
Questions testing right-triangle trig typically give two of the three pieces (an angle and a side, or two sides) and ask for the third. The setup step -- identifying which side is opposite, adjacent, or the hypotenuse relative to the given angle -- is where most errors occur. This simulation practices that identification in every step.