#### Math - 2017-18

# G.8 and *G.8bc - Special Right Triangles and Trig

**G.8** The student will **solve **real-world problems involving right
triangles by using the Pythagorean Theorem and its converse, properties of
special right triangles, and right triangle trigonometry.

G.8The student will solve problems, including practical problems, involving right triangles. This will include applyingb) properties of special right triangles;

c) trigonometric ratios.

**Bloom's Level:** Analyze

*Adopted: 2009*

### BIG IDEAS

- I can determine what size TV to purchase, figure what length
ladder will be needed for a job, and find the shortest route to travel.

- I will calculate the length of a segment and determine a right
angle without directly measuring.

### UNDERSTANDING THE STANDARD

- The Pythagorean Theorem is essential for solving
problems involving right triangles.
- Many historical and algebraic proofs of the
Pythagorean Theorem exist.
- The relationships between the sides and angles
of right triangles are useful in many applied fields.
- Some practical problems can be solved by
choosing an efficient representation of the problem.
- Another formula for the area of a triangle is A = 1/2
*ab*sin*C*. - The ratios of side lengths in similar right triangles
(adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale
factor and depend only on the angle the hypotenuse makes with the adjacent
side, thus justifying the definition and calculation of trigonometric functions
using the ratios of side lengths for similar right triangles.

· The converse of the Pythagorean Theorem can be used to determine if a triangle is a right triangle.

· 45°-45°-90° and 30°-60°-90° triangles are special right triangles because their side lengths can be specified as exact values using radicals rather than decimal approximations.

· The sine of an acute angle in a right triangle is equal to the cosine of its complement.

### ESSENTIALS

G.8_{2}
**Solve** for missing lengths in
geometric figures, using properties of 45-45-90 degree triangles.

· G.8b_{1} **Solve**
for missing lengths in geometric figures, using properties of 45°-45°-90° triangles where
rationalizing denominators may be necessary.

G.8_{3}
**Solve** for missing lengths in
geometric figures using properties of 30-60-90 degree triangles.

· G.8b_{2} **Solve**
for missing lengths in geometric figures, using properties of 30°-60°-90° triangles where
rationalizing denominators may be necessary.

G.8_{4}
**Solve** problems involving
right triangles, using sine, cosine, and tangent ratios.

· G.8c_{1} **Solve**
problems, including practical problems, involving right triangles with missing
side lengths or angle measurements, using sine, cosine, and tangent ratios.

G.8_{6}
**Explain** and **use** the relationship between the sine
and cosine of complementary angles.

G.8_{5}
**Solve** real-world problems,
using right triangle trig and properties of right triangles.

· G.8abc_{1} **Solve**
problems, including practical problems, using right triangle trigonometry and
properties of special right triangles.

### KEY VOCABULARY

triangle, side, angle, right angle, right
triangle, Pythagorean Theorem, Converse of Pythagorean Theorem, special right
triangle, right triangle trigonometry, length, geometric figure, 45°-45°-90° triangle, 30°-60°-90° triangle, sine, cosine, tangent, grade, hypotenuse, short leg, long
leg, complementary angle, angle of elevation, angle of depression, ratio,
similar, adjacent, adjacent side, opposite, opposite, scale factor, justify,
calculate

*Updated: Oct 27, 2017*