Lesson 42 - Review of Right Triangle Trigonometry

Math 2 Honors - Santowski 1

Lesson 42 - Review of Right Triangle Trigonometry

Math 2 Honors – Santowski


(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


(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


(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°

Examples – Right Triangle Trigonometry

504/21/23 Math SL1 - Santowski

Examples – Right Triangle Trigonometry

604/21/23 Math SL1 - Santowski


(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


(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


(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


a=2.4

c=5.2

b=3.5A

B

C

(E) Cosine Law - Examples

Solve this triangle


(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

04/21/23 Math 2 Honors - Santowski

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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.



(H) Homework

Nelson S6.1

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Lesson 42 - Review of Right Triangle Trigonometry