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Lesson 19 Part 1 Pythagorean Theorem and the Primary Trigonometric Ratios
Key Learning Goals
• I can solve for missing sides and angles in right triangles using Pythagorean Theorem and the primary trigonometric ratios sine, cosine and tangent
• I can identify and use the reciprocal trigonometric ratios cosecant, secant and cotangent
• I can solve real­life problems using the six trigonometric ratios
MINDS ON
Accessing Our Prior Knowledge ­ Pythagorean Theorem
Practice.
Solve for the missing side of each right triangle.
A.
B.
1
MINDS ON
Accessing Our Prior Knowledge ­ The Primary Trigonometric Ratios
There is a proportional relationship between a given angle in a right triangle and the length of the sides that make up that triangle.
We will explore this relationship by examining right triangles with different values of the angle A.
Activity: The Geometer's Sketchpad
ACTION
The Primary Trigonometric Ratios
The three primary trigonometric ratios are sine, cosine and tangent. Each ratio compares the following pairs of sides.
sine = opposite
hypotenuse
cosine = adjacent tangent = opposite
hypotenuse adjacent
SOHCAHTOA
So for Δ ABC we have:
sin A =
sin B =
cos A =
cos B = tan A = tan B =
2
Skill: Solving for a missing side in a right triangle using the primary trigonometric ratios.
Example Find the value of the missing side in each triangle. Round to nearest tenth of a unit.
A.
B.
C.
Skill: Solving for a missing angle in a right triangle using the primary trigonometric ratios.
Example Find the value of the missing angle in each triangle. Round to the nearest degree.
A.
B.
C.
3
Skill: Solving a right triangle using the primary trigonometric ratios.
The Reciprocal Trigonometric Ratios
The reciprocal trigonometric ratios are the reciprocals of the primary trigonometric ratios.
Therefore the ratios are:
cosecant = 1 = hypotenuse
(csc) sine opposite
secant = 1 = hypotenuse
(sec) cosine adjacent
cotangent = 1 = adjacent
(cot)
tangent opposite
4
Practice
For the triangle shown below identify the following ratios:
sin P =
csc P =
cos P =
sec P = tan P =
cot P = Practice
State the reciprocal ratios for triangle DEF.
D
E
F
5
Practice
Given that cotθ = 6 , solve for θ.
5
Practice
Given that cscθ = 8 , solve for θ.
3
6
Assigned Practice
Section 5.1, p. 280­282, #1, 3, 4, 6, 9
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