lesson 42 - review of right triangle trigonometry
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Lesson 42 - Review of Right Triangle Trigonometry. Math 2 Honors – Santowski. (A) Review of Right Triangle Trig. - PowerPoint PPT PresentationTRANSCRIPT
Math 2 Honors - Santowski 1
Lesson 42 - Review of Right Triangle Trigonometry
Math 2 Honors – Santowski
Math 2 Honors - Santowski 2
(A) Review of Right Triangle Trig
Trigonometry is the study and solution of Triangles. Solving a triangle means finding the value of each of its sides and angles. The following terminology and tactics will be important in the solving of triangles.
Pythagorean Theorem (a2+b2=c2). Only for right angle triangles
Sine (sin), Cosecant (csc or 1/sin) Cosine (cos), Secant (sec or 1/cos) Tangent (tan), Cotangent (cot or 1/tan) Right/Oblique triangle
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(A) Review of Right Triangle Trig
In a right triangle, the primary trigonometric ratios (which relate pairs of sides in a ratio to a given reference angle) are as follows:
sine A = opposite side/hypotenuse side & the cosecant A = cscA = h/o cosine A = adjacent side/hypotenuse side & the secant A = secA = h/a tangent A = adjacent side/opposite side & the cotangent A = cotA = a/o
recall SOHCAHTOA as a way of remembering the trig. ratio and its corresponding sides
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(B) Examples – Right Triangle Trigonometry Using the right triangle trig ratios, we can solve for
unknown sides and angles:
ex 1. Find a in ABC if b = 2.8, C = 90°, and A = 35°
ex 2. Find A in ABC if c = 4.5 and a = 3.5 and B = 90°
ex 3. Solve ABC if b = 4, a = 1.5 and B = 90°
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(C) Cosine Law
The Cosine Law states the following: a² = b² + c² - 2bccosA b2 = a2 + c2 - 2accosB c2 = a2 + b2 - 2abcosC
We can use the Cosine Law to work in right and non-right triangles (oblique) in which we know all three sides (SSS) and one in which we know two sides plus the contained angle (SAS).
a
c
bA
B
C
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(D) Law of Cosines:
Have: two sides,included angle
Solve for: missing side
(missing side)2 = (one side)2 + (other side)2 – 2 (one side)(other side) cos(included angle)
c2 = a2 + b2 – 2 a b cos C
C
c
A
a
b
B
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(D) Law of Cosines:
C
c
A
a
b
B
a2 + b2 – c2
2abcos C =
Have: three sides
Solve for: missing angle
Missing Angle Side Opposite Missing Angle
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(F) Examples Cosine Law
We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles:
ex 1. Find c in CDE if C = 56°, d = 4.7 and e = 8.5
ex 2. Find G in GHJ if h = 5.9, g = 9.2 and j = 8.1
ex 3. Solve CDE if D = 49°, e = 3.7 and c = 5.1
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(G) Review of the Sine Law
If we have a non right triangle, we cannot use the primary trig ratios, so we must explore new trigonometric relationships.
One such relationship is called the Sine Law which states the following:
AB
C
sinsinsin
OR sinsinsin c
C
b
B
a
A
C
c
B
b
A
a
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(G) Law of Sines: Solve for Sides
C
c
A
a
b
B
Have: two angles, one side opposite one of the given angles
Solve for: missing side opposite the other given angle
Missing Side
a sin A
= b sin B
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(G) Law of Sines: Solve for Angles
C
c
A
a
b
B
Have: two sides and one of the opposite angles
Solve for: missing angle opposite the other given angle
Missing Angle a sin A
= b sin B
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(H) Examples Sine Law
We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles:
ex 4. Find A in ABC if a = 10.4, c = 12.8 and C = 75°
ex 5. Find a in ABC if A = 84°, B = 36°, and b = 3.9
ex 6. Solve EFG if E = 82°, e = 11.8, and F = 25°
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, the Sine Law cannot be used.
04/21/23 Math 2 Honors - Santowski