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Trigonometry. Success Criteria

o I can solve right triangle problems using the Pythagorean Theorem;

o I can solve right triangle problems using the three primary trigonometric ratios sine, cosine, and tangent;

o I can solve right triangle problems for an angle in degrees using inverse sine, inverse cosine, and inverse tangent

o I can solve oblique (non-right) triangles using the Sine Law;o I can solve oblique (non-right) triangles using the Cosine

Law;Fact 1: A triangle can be formed from three non-collinear points in a plane. If the points are collinear, then they do not form a triangle.

Fact 2: For any triangle, the internal angles sum to 180 degrees.

Fact 3. A triangle that has a 90 degree internal angle is called a right triangle. (Also known as a right-angled triangle.)

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Fact 4. An oblique angle is any angle that is not a right angle or multiples of a right angle.

Fact 5. An acute angle is an angle between zero degrees and ninety degrees.

Fact 6. An obtuse angle is an angle between 90 degrees and 180 degrees.

Fact 7. To solve a triangle means to find the measures of all side lengths and internal angles.

Fact 8. Equilateral triangles have all sides the same length.

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Fact 9. Isosceles triangles have two sides the same length.

Fact 10. Scalene triangles have no sides of equal length.

Fact 11. The Pythagorean Theorem only applies to right triangles.

Fact 12. The three primary trigonometry ratios are sine, cosine, and tangent. Each of them is a ratio comparison of side lengths. The phrase “SOH CAH TOA” is a memory aid.

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Fact 13. Sine Law: For any triangle ABC, asin A

= bsinB

= csinC . In

practice, we only use two of these ratios at any time to solve problems. For example:

asin A

= bsinB

Example:

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Example 2.

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Hint: In practice, the Sine Law can also be used as:

(a)(sin B) = (b) (sin A)

Fact 14: The Cosine Law:

For any triangle ABC, a2=b2+c2−[2bc (CosA )].

So, if we know the values of sides b, c, and the angle A, then we can find the measure of the missing third side a.

Similarly,b2=a2+c2−[2ac (CosB )], which we can use to find length b; and

c2=a2+b2−[2ab (CosC )], which we can use to find length c.

Thus, the Cosine Laws allow us the ability to determine the length of the missing third side whenever we know the lengths of two sides and the contained angled between the two known side lengths.Hint: Watch out for negative sign errors! Use brackets when substituting values into equations.

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Fact 15. If we know the measures of all three sides of any triangle, then the Cosine Laws can be rearranged to allow us to solve for any missing internal angle A, B, or C. The formulas are:

cosA=b2+c2−a2

2bc ; or sB= a

2+c2−b2

2ac ; or cosC=a

2+b2−c2

2ab .

Fact 16.

Fact 17. An angle of elevation is also called an angle of incline or angle of inclination. Inclines angle up from a point, while declines angle down from a point.

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Fact 18. The x-axis and y-axis form quadrants on a plane.

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Fact 19. Supplementary angles sum to 180 degrees. So, angle A + supplementary angle A’ = 180 degrees.

Question: Compare the sinA to sinA’

Question: Compare the cosA to cosA’

Question: Compare the tanA to tanA’

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Fact 20. Points on a plane form right angles to the x-axis.

Question: What are the coordinates of point P, shown above?

Question: How far is point P from the origin? Show the calculations

Question: What angle does the line segment OP make with the x-axis?

Question: What is the measure of the supplementary angle to angle BOP?

Question: What is the sin of angle BOP?

Question: What is the cos of angle BOP?

Question: What is the tan of angle BOP?

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Example:

Fact 21.

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Hint:

Students need to practice problem solving skills.