Download Trigonometry Lesson 1: Primary Trigonometric Ratios

Trigonometry Lesson 1: Primary Trigonometric Ratios
Todays Objectives:
 Students will be able to develop and apply the primary trigonometric ratios (sine,
cosine, tangent) to solve problems that involve right triangles, including:
o Identify the hypotenuse of a right triangle and the opposite and adjacent
sides for a given acute angle in the triangle
o Explain the relationships between similar right triangles and the
definitions of the primary trigonometric ratios
Let’s begin with a short review of some procedures that are common in trigonometry
problems before we discuss the primary trigonometric ratios.
Right Triangles
 A _______ triangle has a right (90º) angle. The other two angles are each
_______(between 0º and 90º). The side _______ the right angles is the longest
side, and is called the ____________. The two sides__________ to the right
angle are called _____ of the right triangle. The word adjacent means
“_________”.
Pythagorean Theorem
 In any right triangle, the ________ of the hypotenuse is equal to the _____ of
the squares of the other two _______ (legs). This can be illustrated as follows:
Note: __________ are often labeled with _________ letters, and the sides__________
the vertices are labeled with the corresponding _______ case letters. We normally label
the hypotenuse as ____.
Example)
Solve for the unknown side length, x.
Solution:
Sum of the Angles in a Triangle
 In any triangle, the sum of the measures of the three angles is always equal to
180º
Example)
Determine the measure of angle XYZ. (When the name of the angle is given as three
letters, the middle letter represents the vertex of the angle)
Sine, Cosine and Tangent Ratios
The three primary trigonometric _______ describe the ratios of the different ______ in
a right triangle.These ratios use one of the _______ angles as a point of reference. The
90º angle is ______ used. In the following illustration, the ratios are described relative
to _________. Notice that the abbreviations for sine, cosine, and tangent are ____, ____,
and ____.
SOHCAHTOA
You can use the acronym SOH-CAH-TOA to remember these ratios
Example
Determine the three primary trigonometric ratios from angle θ
Solution: first, find the unknown side x
Now, write the three ratios
Trigonometric Ratios and Similar Triangles
__________triangles are triangles in which the corresponding _______ have the same
_________. The corresponding _____ in similar triangles are ________________. One
way of constructing similar right triangles is shown in the given diagram below.
Angles with the same markings have the same measure. Three similar triangles have
been formed: ∆AB1C1, ∆AB2C2, ∆AB3C3
Using each of these triangles, consider the ____ ratio for __________. Because the sides
are proportional, the sin ratios using each of the three similar triangles are _______.
Sin A = opposite/hypotenuse
= B1C1/AB1
=B2C2/AB2
= B3C3/AB3
Example)
Complete the following steps for the three similar triangles formed in the given diagram.
1) State the tangent of angle θ using the labels of the sides
2) Use a metric ruler to measure the side lengths for each triangle, and give an
estimate of the value of tan θ to the nearest hundredth
3) Calculate the value of angle θ
Note: Calculators need to be set to Degree Mode, not Radian Mode. Inverse functions
are available by using the shift or 2nd function key on your calculator.