circle.
If students have trouble remembering these special shortcuts, encourage them to use Pythagorean’s Theorem and
simplify the answer. The resulting answer will equal the shortcut.
In-class Activity!Separate your class into six to ten groups. Write six to ten numbers on the board, one for each
group. Instruct the groups to draw an isosceles right triangle with legs of the given length. Have the groups solve for
the hypotenuse and share with the remaining groups.
What is the Connection? In any right isosceles triangle, if a leg has value ofx, then by Pythagorean’s Theorem,
x^2 +x^2 =h^2. Adding like terms, you get 2x^2 =h^2. To solve for the value of the hypotenuse, you must square root
both sides, leaving the equationx
√
2 =h. Therefore, the length of the hypotenuse in ANY right isosceles triangle
is equal to leg
√
2
In any 30− 60 −90 triangle, the relationship between the segments is as follows: Let the smallest leg have the value
ofd. The hypotenuse will always have length 2dand the other leg will always have lengthd
√
- This again can be
proved using Pythagorean’s Theorem.
Tangent Ratios
Pacing:This lesson should take one class period
Goal:This lesson introduces the first trigonometric function, the tangent ratio. The tangent seems to be the most
natural for students to understand, as opposed to the sine or cosine functions. Tangent ratios occur in many careers,
from construction to machine operators.
Check Your Tech!If you are using calculators for the tangent (TAN), cosine (COS), and sine (SIN) functions, be
sure to do a “Mode Check.” Have each student check to ensure their calculator is set to degrees (DEG) instead of
radians (RAD). Having a calculator in radians will provide incorrect answers and students at this level do not know
what radians are to correct their answers.
Vocabulary Connection!Students must understandadjacentandoppositeto be successful with trigonometric ratios.
They have already had experience with adjacent in previous chapters. Begin by reviewing such vocabulary as
adjacent angles and adjacent sides in regards to parallel lines and transversals.
Beginning Activity!Once the class has reviewed the termadjacent,offer students these triangles. Ask students to
write the termsadjacentandoppositeabove the appropriate legs and label the hypotenuse. Always stress that the
information stems from the given angle (not the 90 degree)!
Extension!When discussing tangent values of common angles, such as 30◦, 45 ◦,and 60◦, review how to rationalize
the denominator with your students. This concept should have been presented in Algebra 1. While not as common
with the increased use of technology, most standardized test questions will present answers in completely simplified
form. For example, the tangent( 30 ◦) =√^1
3
. The objective of rationalizing a denominator is to clear it of decimals,
radicals, and complex values. To do so, multiply the fraction by a value of 1, in this case,
√
√^3
3
. The new expression
1.8. Right Triangle Trigonometry