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TF.01.1 - Review of Right TF.01.1 - Review of Right Triangle Trigonometry, Sine Law Triangle Trigonometry, Sine Law

and Cosine Lawand Cosine Law

MCR3U – SantowskiMCR3U – Santowski

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(A) Review of Right Triangles(A) Review of Right Triangles

In a right triangle, the primary trigonometric ratios (which In a right triangle, the primary trigonometric ratios (which relate pairs of sides in a ratio to a given reference angle) are as relate pairs of sides in a ratio to a given reference angle) are as follows:follows:

sine A = opposite side/hypotenuse side sine A = opposite side/hypotenuse side cosine A = adjacent side/hypotenuse sidecosine A = adjacent side/hypotenuse side tangent A = adjacent side/opposite sidetangent A = adjacent side/opposite side recall SOHCAHTOA as a way of remembering the trig. ratio and recall SOHCAHTOA as a way of remembering the trig. ratio and

its corresponding sidesits corresponding sides

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(B) Examples – Right Triangle (B) Examples – Right Triangle TrigonometryTrigonometry

Using the right triangle trig ratios, we can solve Using the right triangle trig ratios, we can solve for unknown sides and angles:for unknown sides and angles:

ex 1. Find ex 1. Find aa in ABC if in ABC if bb = 2.8, = 2.8, CC = 90°, and = 90°, and AA = = 35°35°

ex 2. Find ex 2. Find AA in ABC if in ABC if cc = 4.5 and = 4.5 and aa = 3.5 and = 3.5 and BB = = 90°90°

ex 3. Solve ABC if ex 3. Solve ABC if bb = 4, = 4, aa = 1.5 and = 1.5 and BB = 90° = 90°

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(C) Review of the Sine Law(C) Review of the Sine Law

If we have an acute triangle, we cannot use the primary trig ratios, so we If we have an acute triangle, we cannot use the primary trig ratios, so we must explore new trigonometric relationships. must explore new trigonometric relationships.

One such relationship is called the Sine Law which states the following:One such relationship is called the Sine Law which states the following:

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(D) Examples Sine Law(D) Examples Sine Law

We can use these new trigonometric relationships in solving We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles:for unknown sides and angles in acute triangles:

ex 4. Find ex 4. Find AA in ABC if in ABC if aa = 10.4, = 10.4, cc = 12.8 and = 12.8 and CC = 75° = 75°

ex 5. Find ex 5. Find aa in ABC if in ABC if AA = 84°, = 84°, BB = 36°, and = 36°, and bb = 3.9 = 3.9

ex 6. Solve EFG if E = 82°, ex 6. Solve EFG if E = 82°, ee = 11.8, and = 11.8, and FF = 25° = 25°

There is one limitation on the Sine Law, in that it can only There is one limitation on the Sine Law, in that it can only be applied if a side and its opposite angle is known. If not, be applied if a side and its opposite angle is known. If not, the Sine Law cannot be used.the Sine Law cannot be used.

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(E) Cosine Law(E) Cosine Law

The Cosine Law states the following: The Cosine Law states the following: aa² = ² = bb² + ² + cc² - 2² - 2bcbcCosCosAA

We can use the Cosine Law to work in triangles in which the We can use the Cosine Law to work in triangles in which the Sine Law does not work - triangles in which we know all three Sine Law does not work - triangles in which we know all three sides and one in which we know two sides plus the contained sides and one in which we know two sides plus the contained angle.angle.

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(F) Examples Cosine Law(F) Examples Cosine Law

We can use these new trigonometric relationships We can use these new trigonometric relationships in solving for unknown sides and angles in acute in solving for unknown sides and angles in acute triangles:triangles:

ex 7. Find ex 7. Find cc in CDE if in CDE if CC = 56°, = 56°, dd = 4.7 and = 4.7 and ee = 8.5 = 8.5

ex 8. Find ex 8. Find GG in GHJ if in GHJ if hh = 5.9, = 5.9, gg = 9.2 and = 9.2 and jj = 8.1 = 8.1

ex 9. Solve CDE if ex 9. Solve CDE if DD = 49°, = 49°, ee = 3.7 and = 3.7 and cc = 5.1 = 5.1

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(G) Links(G) Links

For help with right triangle trig: For help with right triangle trig: http://id.mind.net/~zona/mmts/trigonometryRealhttp://id.mind.net/~zona/mmts/trigonometryRealms/introduction/rightTriangle/trigRightTriangle.htms/introduction/rightTriangle/trigRightTriangle.htmlml

For help with the Sine LawFor help with the Sine Law http://www.themathpage.com/aTrig/law-of-sines.http://www.themathpage.com/aTrig/law-of-sines.

htmhtm

For help with the Cosine LawFor help with the Cosine Law http://www.themathpage.com/aTrig/law-of-cosinehttp://www.themathpage.com/aTrig/law-of-cosine

s.htms.htm

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(H) Homework(H) Homework

AW TextAW Text Page 247, Q1-5Page 247, Q1-5 Page 249, Q,5,6Page 249, Q,5,6 Page 252, Q,5,6Page 252, Q,5,6

Nelson Text, p497-8, Q11-14Nelson Text, p497-8, Q11-14