Montana Math MT.CORE.GEOM.MEAS.3

MT.CORE.GEOM.MEAS.3 is a cluster heading. These are the individual standards under it.

• MT.CORE.GEOM.MEAS.3.a

  Deriving and applying the trigonometric ratios in special right triangles, and

• MT.CORE.GEOM.MEAS.3.b

  Using trigonometric ratios and the Pythagorean Theorem to solve right triangles.

01

What Students Need to Do

Students compare similar right triangles to see why the side ratios stay the same for a given angle. They find exact ratios for 30°, 45°, and 60°, then use trig ratios and the Pythagorean theorem to find missing sides and angles.

02

What Mastery Looks Like

A student can draw an altitude in an equilateral triangle to find the side lengths of a 30°-60°-90° triangle. Given an angle and a side in another right triangle, they choose the correct ratio, solve for missing lengths, and check them with the Pythagorean theorem.

03

Common Misconceptions

Students may label sides opposite and adjacent without checking which angle they are using. They may treat sine as a side length rather than a ratio, or assume a calculator result for 30° is exact when an exact fraction or radical is needed.

04

How to Assess It

Exit ticket: A 30°-60°-90° triangle has a short leg of 5 cm. Find the other two sides and tan 30°, then use the Pythagorean theorem to check the side lengths.

1. 01

   Cut two different-sized paper squares along a diagonal, then measure and compare the side ratios in the resulting 45° triangles.

2. 02

   Draw an altitude in an equilateral triangle and explain how its half-base helps you find sin 30°, cos 30°, and tan 30°.

3. 03

   Give pairs four index cards showing right triangles with one angle and one side; have them solve missing sides and check with the Pythagorean theorem.

4. 04

   Measure 10 meters from a flagpole, use a phone inclinometer to find the angle to its top, and estimate its height, including eye height.

• MT.PLUS.ALG.TRIG.5

  Apply the Law of Sines and the Law of Cosines to find unknown measurements in non-right triangles.

• MT.PLUS.ALG.TRIG.1.b

  Defining the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine.