Montana Math MT.CORE.GEOM.MEAS.3.a
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: Deriving and applying the trigonometric ratios in special right triangles, and
2026 Montana Mathematics Content Standards · Geometric Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students show why similar right triangles have the same side ratios for a given angle. They find the exact ratios for 30°, 45°, and 60°, then use them to find missing sides.
What Mastery Looks Like
- A student can split a square or equilateral triangle to work out the special triangle side lengths. They can identify opposite, adjacent, and hypotenuse from a marked angle and use the right ratio to find a missing length.
Common Misconceptions
- Students may treat opposite and adjacent as fixed sides, even when the marked angle changes. They may swap sine and cosine, or think the side opposite 30° is the longer leg.
How to Assess It
- Give students a 30°-60°-90° triangle with hypotenuse 12. Ask them to find both legs, write sin 30°, cos 30°, and tan 30°, and explain why the ratios stay the same in a larger triangle.
Lesson moves
Ways to Teach It
Cut a unit square diagonally and fold a side-2 equilateral paper triangle along its altitude, then use the Pythagorean theorem to label all sides.
Have students write why sin 30° stays the same when a triangle with hypotenuse 2 is enlarged to hypotenuse 10.
Give pairs cards showing special right triangles with one missing side; partners match each card to an exact ratio and solve.
Sketch a roof with a 30° slope and an 8-foot rise, then use sin 30° to find the rafter length.
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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.b
Using trigonometric ratios and the Pythagorean Theorem to solve 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.CORE.GEOM.TRANS.3.a
Verifying that two triangles are similar if, and only if, corresponding pairs of sides are proportional and corresponding pairs of angles are congruent, and
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