Name: _________________________________________________________ 14-7 Trig and Area on the plane Review Geometry Pd. ______ Date: __________

14-7: Trigonometry and area on the coordinate review

PART 1-TRIGConcept 1: Trig Ratios - SohCahToa

Key Ideas/Tips Regents Examples/ MY NOTESTrigonometric Ratios – SOHCAHTOA ****PUT CALCULATOR IN DEGREES***- Used to solve for sides or to solve for angles

- When solving for a side remember to use: sine, cosine, or tangent and cross multiply!

- When solving for an angle remember to use: sine inverse, cosine inverse, or tangent inverse

ex) sin-1(sinθ) = sin-1(45 )

sin-1(sinθ) = sin-1(45 )

θ = 53.1°*Angle of Elevation – angle formed from ground up

*Angle of Depression – angle formed from eye-level down

The The diagram below shows two

similar triangles. If tanθ=1013 , what is

the value of x to the nearest tenth?

(1) 4.4 (2) 4.3 (3) 7.3 (4) 7.2

Concept 2: Law of SinesKey Ideas/Tips Regents Examples/ MY NOTES

Law of SinesTip: Read carefully, if it is NOT a right triangle we

can’t use SOHCAHTOA!o

o The Law of Sines can be used to find... you have 2 sides and 2 angles of a

triangle, including the unknown.

Must find the “pairs” (the angles and sides across from each other)

In the What is the distance from campsite A to campsite B, to the nearest yard?1) 1,4692) 1,1503) 2,140

4) 2,141

Concept 3: Law of CosinesKey Ideas/Tips Regents Examples/ MY NOTES

Law of Cosines

**NEED TO USE THE ANGLE OPPOSITE “a2.”

Law of Cosines can be used if... you have 3 sides and 1 angle of a triangle, including the unknown.

Find the size of thesmallest angle to thenearest degree.

Concept 4: CofunctionsKey Ideas/Tips Regents Examples/ MY NOTES

Cofunctionssin θ=¿cos¿¿

Try it! Write a trig function equivalent to the following, but with an angle value less than 45°

sin 78 ¿cos¿ = cos ( )

When asked to explain :

When two angles are complementary, the value of the sine of one angle is equal to the value of the cosine of the other angle.

2. In , the complement of is . Which statement is always true?

1)2)3)4)

Find

2. Find the value of R that will make the equation true when . Explain your

answer.

Self-Assess: Where should you start?Rank the following concepts with 1-4, where 1 = “I feel the most confident with this topic!” And 4 = “I feel the

least comfortable with this topic.”

Concept 1 Concept 2 Concept 3 Concept 4

Concept 1: Right Triangle Trig Ratios SOHCAHTOA – CALCULATOR in DEGREE MODE!!!!!!

1. Which ratio represents the cosine of angle A in the right triangle below?1)

2)

3)

4)

2. In triangle MCT, the measure of , , , and . Which ratio represents the sine of ? (Hint: DRAW IT!)

1)

2)

3)

4)

Complete both Column A AND Column B:

Column A Column B

3. Solve for the value of x, to the nearest tenth. 4. Solve for the m∡ B to the nearest hundredth of a degree

5. A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below

If the angle of elevation from the tip of the shadow to the top of the tree is 32° , what is the height of the tree to the nearest tenth of a foot?

6. The diagram below shows the path a bird flies from the top of a 9.5 foot tall sunflower to a point on the ground 5 feet from the base of the sunflower.

To the nearest tenth of a degree what is the measure of angle x?

Concept 2: Law of Sines***IMPORTANT! You will NOT BE GIVEN these equations on the test. You have to memorize each.

7. Given triangle ABC, m∡ A=97 ° , side AB=19 ft , side BC=22 ft∧side AC=9 ft . Find the measurement of ∡C to the nearest degree.

x

16

8. Given triangle¿, ∠D=22 °, ∠F=91 °, DF=16.55, and EF=6.74, find DE to the nearest whole number.

Concept 3: Law of Cosines9. Find the largest angle, to the nearest tenth of a degree, of a triangle whose sides are 9, 12 and 18

10. The playground at a day-care center has a triangular-shaped sandbox. Two of the sides measure 25 feet

and 18.5 feet and form an included angle of 52°. Find the length of the third side of the sandbox to the

nearest tenth of a foot.

a. Which law do we use here? ___________________________

b. Solve for the third side of the sandbox.

Concept 4: CofunctionsSolve the following problems using your knowledge of cofunctions:

11. If , find x.

12. Explain how you got your answer.

13. If Cos A = 2x + 57 and Sin B = 5x in right triangle ABC, with a right angle at C, find the value of x . Explain how you got your answer.

14. Given: Right triangle ABC with right angle at C. If increases, does increase or decrease? Explain why.

PART 2-Area on the GridConcept 1: Area of Overlapping polygons

Key Ideas/Tips Regents Examples/ MY NOTESUnion (∪¿- Add area of each shape minus overlap

Watch out! Careful when calculating area triangles are tricky, identify the height and base first!

Determine the area of the ΔABC ∪ ΔDEF.

Intersection¿)- Area of JUST the overlap Determine the area of the ΔABC∩ ΔDEF.

Concept 2: Area of slanted PolygonsKey Ideas/Tips Regents Examples/ MY NOTES

Area of Slanted Polygons Draw a “box” around slanted polygon. Label each region with Roman numeral (including

the missing region).

Triangle RST is graphed on the set of axes below.How many square units are in the area of ?

A∪B

A ∩ B

Calculate the area of each region and subtract form the area of the “box”

Watch out! Careful when calculating area what shapes have you made with the box? Rectangle? Square? Triangle? If the

shapes are no these, REDRAW BOX.

Concept 3: Shortest Distance from a point to a lineKey Ideas/Tips Regents Examples/ MY NOTES

Finding the shortest distance between a point and a line.

Memorize!

Find the shortest distance from the point (-15, 2.5) to the line 5x+6y=30 to the nearest tenth.

Self-assesses For success – Where should you start? Rank the following concepts 1-3 where 1= “ I feel most confident with this topic” 3=” I feel the least comfortable with this topic”

Area of overlapping polygons

Area of slanted Polygons

Shortest Distance from a point to a line

Concept 1 : Area of overlapping polygons

1. In the accompanying diagram, right triangle ABC is inscribed in a circle. BA is the diameter, BC =6 cm, AC = 8 cm. Find the exact area of the shaded region. Leave your answer in terms of π

Shortest distance=

2. Determine the area of the ΔABC∪ ΔDEF to the nearest tenth..

3. Polygon A is a square and Figure B is a circle. Calculate the area of A U B. Round your answer to the nearest inch.

4. A designer created the logo shown below. The logo consists of a square and four quarter circles of equal size. Express in terms of π, the exact area in square inches of the shaded region.

Concept 2 : Slanted polygons 5. Determine the area of the following quadrilateral

6. Determine the area of a triangle ABC, whose vertices are A(-3,1) , B(1,3) and C(3,-2).

Concept 3 : Shortest distance from a point to a line

7. Determine the shortest distance between the given line of slope -2 and the given point, leave answer in radical form.

8. Determine the shortest distance between a point and a line: y=−12

x+2 and point (4,-2) round to the nearest

whole number.