Montana Math MT.CORE.GEOM.MEAS.3.b
The Standard
- MT.CORE.GEOM.MEAS.3
- Use similarity to explore and define the sine ratio, cosine ratio, and tangent ratio in terms of right triangles by: Deriving and applying the trigonometric ratios in special right triangles, and Using trigonometric ratios and the Pythagorean Theorem to solve right triangles.
Use similarity to explore and define the sine ratio, cosine ratio, and tangent ratio in terms of right triangles by: Using trigonometric ratios and the Pythagorean Theorem to solve right triangles.
2026 Montana Mathematics Content Standards · Geometric Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students compare scaled right triangles to see that side ratios stay the same for a given angle. They define sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. They use these ratios and the Pythagorean theorem to find missing sides and angles.
What Mastery Looks Like
- Given one acute angle and one side, students choose a ratio and find a missing side. Given two sides, they find an unknown angle and use the Pythagorean theorem to check the third side.
Common Misconceptions
- Students may label opposite and adjacent without checking which angle is marked. They may put the hypotenuse in a tangent ratio, or forget to take the square root after adding squares.
How to Assess It
- Exit ticket: A right triangle has legs of 6 cm and 8 cm, with angle A opposite the 6 cm leg. Find the hypotenuse, angle A, and tan A, then say what happens to tan A if both legs double.
Lesson moves
Ways to Teach It
Cut two similar right triangles from grid paper, mark the same angle, and compare their opposite-to-hypotenuse, adjacent-to-hypotenuse, and opposite-to-adjacent ratios.
Draw two scaled right triangles and write why their sine, cosine, and tangent values match for the same marked angle.
Give pairs right-triangle cards showing one angle and one side; have them find both missing sides using a trig ratio and the Pythagorean theorem.
Measure the distance to a flagpole and sight its top with a phone inclinometer, then use tangent and eye height to estimate its height.
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Related Standards
- MT.PLUS.ALG.TRIG.5
Apply the Law of Sines and the Law of Cosines to find unknown measurements in non-right triangles.
- MT.CORE.GEOM.MEAS.3
Use similarity to explore and define the sine ratio, cosine ratio, and tangent ratio in terms of right triangles by: Deriving and applying the trigonometric rat...
- MT.8.G.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangle problems, including problems in context in two and three dimensions. This stan...
- MT.CORE.GEOM.MEAS.3.a
Deriving and applying the trigonometric ratios in special right triangles, and
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