Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Build the tangent ratio.
- Use inverse tangent in degree mode.
- Check an answer and its complement.
Before you start
Right triangles, tangent, and calculator degree mode.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A line in a model rises 3 cm over a horizontal run of 4 cm. Find its acute angle above horizontal and its angle from vertical.
Why this math matters
A forward trig function converts an angle to a ratio. An inverse function reverses that question within a specified output interval. This distinction is useful when dimensions are known but inclination is not.
Set up the model
A useful answer starts with clear assumptions:
- The horizontal and vertical measurements are perpendicular.
- The line rises to the right, and its inclination is between 0° and 90°.
- The model describes geometry only, not a construction specification.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recover an angle from a rise and run
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A line in a model rises 3 cm over a horizontal run of 4 cm. Find its acute angle above horizontal and its angle from vertical.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Relate the given sides
tan θ = 3/4 = 0.75
Vertical rise is opposite the horizontal reference angle, and horizontal run is adjacent.
Undo the tangent
θ = arctan(0.75) ≈ 36.87°
The calculator returns a principal angle. The stated acute-angle condition makes that returned angle the required one.
Check and complement
tan(36.87°) ≈ 0.75; 90° − θ ≈ 53.13°
Substituting the angle checks the ratio. The angle from vertical fills the remainder of the right angle.
The result
The angle is approximately 36.87° above horizontal, or 53.13° from vertical.
A tangent ratio determines angles only after a range or quadrant is specified. Adding 180° gives the same tangent but not the same directed ray.
Common mistakes to catch
- The reciprocal 1/0.75 is not the inverse-tangent angle.
- Radians mode returns about 0.644 radians, which must not be labelled degrees.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A positive rise equals its horizontal run. Find the acute inclination.
Show a hint
Tangent equals 1.
Reveal answer and explanation
45°
arctan(1) is 45°, making an isosceles right triangle.
Practice 2
An acute angle has sine 1/2. Find the angle and its complementary angle.
Show a hint
Use inverse sine, then subtract from 90°.
Reveal answer and explanation
30° and 60°
Within the acute range, sin30° = 1/2 uniquely identifies the angle.
Take the idea with you
State the allowed angle range before using any inverse trigonometric function.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use the unit circle beyond acute angles
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