Use similarity to explain why sine, cosine and tangent depend on an acute angle.
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Try
A 5–12–13 right triangle gives sin θ=5/13, cos θ=12/13 and tan θ=5/12 for θ at the left vertex in the diagram. The labels opposite and adjacent depend on θ; the hypotenuse does not.
Predict
Predict what happens to these ratios when the triangle doubles in size. All side lengths double and the ratios remain unchanged. For an acute angle θ, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These definitions here concern a right triangle, not an arbitrary triangle.
Experiment
Choose the other acute angle in the same diagram and relabel the legs. Its sine is 12/13, while its cosine is 5/13. Rotate your paper: the labels should follow the angle rather than the page’s horizontal direction. Check that both acute angles together sum to 90°.
A 5–12–13 right triangle drawn to scale. Relative to θ at the left vertex, opposite=5, adjacent=12, hypotenuse=13.
Understand
Worked argument: a right-triangle model of a ramp has horizontal run 12 and rise 5. Its length is 13 by Pythagoras. The slope rise/run is 5/12, equal to tangent of the inclination; it is not the sine, whose denominator is 13. Two right triangles with the same acute angle are similar by AA, so each corresponding ratio is independent of scale.
Invalid height estimate: an observer with eye level 1.6 m uses horizontal distance times tangent and calls the result the total mast height. That product gives height above the eye’s horizontal line. On level ground, the eye height must also be added. State the observation geometry before calculating.
Apply
For each application, draw the right triangle, name the target angle, and label measured versus inferred quantities. Numerical ratio checks do not establish that a real scene has level ground or a vertical object. The practice uses explicit idealized assumptions so the mathematics is inspectable.
Continue
Depth: inverse trigonometric functions recover an acute angle from an appropriate ratio. The later unit-circle definition extends sine and cosine beyond acute angles and explains signs and periodicity. Do not treat this right-triangle unit as coverage of the complete trigonometry curriculum. Explore the angle and sine/cosine investigation for a visual bridge to the later unit-circle definition.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
In a right triangle, the side opposite acute angle θ is 5 and the hypotenuse is 13. Find sin θ.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A vertical mast stands on horizontal ground. From a ground-level observation point 24 m horizontally from its base, θ is the angle of elevation from the horizontal to the mast's top. If tan θ=7/24, find the mast height in metres.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For acute θ in a right triangle, the adjacent leg is:
If acute angles α and β are complementary in a right triangle, which identity holds?
Explain why two right triangles with the same acute angle have equal opposite/hypotenuse ratios.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive sin²θ+cos²θ=1 for an acute angle in a right triangle.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
