Honors Precalc (Caron) Final Notes (6.5, 9.1, 9.2)

A helicopter hovers 800 ft directly above a small island. From the helicopter the pilot takes a sighting to point P directly ashore on the mainland, at the water's edge. If the angle of depression is 35°, how far off the coast is the island?

90 - 35 = 55 tan 55° = x/800 x = 800 tan 55° or 1150 ft

Find the area of an acute triangle given that it has a base of 12 cm, a side of 3 cm, and the included angle is 60°

A = (1/2)ab sin θ A = (1/2)(3)(12) sin 60° A = 9√3 cm²

Find sin β and cos β if β is an acute angle with adjacent 1 and opposite 3.

Pythagorean theorem: 3² + 1² = x², hypotenuse = √10 sin β = 3√10/10 cos β = √10/10

How can you remember the formulas for cos, sin, and tan?

SOHCAHTOA

segment

a region of a circle bounded by an arc of the circle and its chord

If a and b are lengths of two sides of a triangle and θ is the area included between those two sides, then what is the area of the triangle given by?

area = (1/2)ab sin θ

Given a segment with a central angle of 3π/5 with sides of 2 cm each and a separate shaded segment, compute the area of the segment in an exact expression involving π and a calculator approximation rounded to two decimal places.

area of segment = area of sector - area of triangle OPQ area of sector = (1/2)r²θ = (1/2)(2²)(3π/5) = 6π/5 cm² area of triangle = (1/2)(2)(2) sin 3π/5 = 2 sin 3π/5 cm² area of segment = (6π/5) - (2 sin 3π/5) = 1.87 cm²

Describe the Law of Cosines

a² = b² + c² - 2bc cos A b² = c² + a² - 2ca cos B c² = a² + b² - 2ab cos C

Given a 30-60-90 triangle with a hypotenuse of 100 cm and an adjacent side of x use one of the trigonometric functions to find x.

cos 30° = adjacent/hypotenuse = x/100 x = 100 cos 30° x = 100 * √3/2 = 50√3 cm

In triangle ABC the sides are a = 3 units, b = 5 units, and c = 7 units. Find the angles.

cos A = (b² + c² - a²)/2bc = (5² + 7² - 3²)/2(5)(7) = 65/70 = 13/14 cos⁻¹ (13/14) = 21.8° cos B = (c² + a² - b²)/2ca = (7² + 3² - 5²)/2(7)(3) = 33/42 = 11/14 cos⁻¹ (11/14) = 38.2° 180 - (21.8 + 38.2) = 120°

Describe the following for a triangle with hypotenuse 5, opposite 3, and adjacent 4: . cos . sin . tan . sec . csc . cot

. 4/5 . 3/5 . 3/4 . 5/4 . 5/3 . 4/3

State the formulas for the following trigonometric functions of an acute angle: . cos θ . sin θ . tan θ . sec θ . csc θ . cot θ

. adjacent/hypotenuse . opposite/hypotenuse . opposite/adjacent . hypotenuse/adjacent . hypotenuse/opposite . adjacent/opposite

Describe the following sides of a triangle: . hypotenuse . opposite . adjacent

. long side of triangle . side opposite angle θ . side adjacent to angle θ

Two satellite-tracking stations, located at points A and B in a desert, are 200 miles apart. At a prearranged time both stations measure the angle of elevation of a satellite as it crosses the vertical plane containing A and B. This means that A, B, and S lie in a plane perpendicular to the ground. If the angles of elevation from A and from B are α and β, express the altitude h of the satellite in terms of α and β.

cot β = (CA + 200)/h h cot β = CA + 200 cot α = CA/h; CA = h cot α h cot β = h cot α + 200 h cot β - h cot α = 200 h(cot β - cot α) = 200 h = 200/(cot β - cot α) miles

If θ is an acute angle and sin θ = t, find express the other five trigonometric functions as function of t

csc θ = 1/t t² + cos² θ = 1 cos θ = √(1-t²) sec θ = 1/√(1-t²) tan θ = t/√(1-t²) cot θ √(1-t²)/t

Given a regular pentagon inscribed in a circle of radius 2 in, find the area of the pentagon.

in a regular n-sided polygon, the central angle is 360°/n. central angle of BOA = 72° area of triangle BOA = (1/2)(2)(2) sin 72° = 2 sin 72° in² *5 for area of pentagon = 10 sin 72° in²

Describe the Law of Sines

in any triangle, the ration of the sine of an angle to the length of its opposite side is constant: sin A/a = sin B/b = sin C/c or a/sin A = b/sin B = c/sin C

Given a 30-60-90 triangle with a hypotenuse of 100 cm and an opposite side of y, use one of the trigonometric functions to find y.

sin 30° = opposite/hypotenuse = y/100 y = 100 sin 30° y = 100 * (1/2) y = 50 cm

Given a triangle with side x opposite angle 135° and side 20 cm opposite angle 30°°, find the length x.

sin 30°/20 = sin 135°/x x sin 30° = 20 sin 135° x = (20 sin 135°)/sin 30° = 20(√2/2)/(1/2) = 20√2 cm

What is an example of the right-triangle identity sin θ/cos θ = tan θ?

sin 45/cos 45 = tan 45

What are examples of the right-triangle identities sin (90°-θ) = cos θ (sin (π/2-θ) = cos θ) and cos (90° - θ) = sin θ (cos(π/2-θ) = sin θ)?

sin 70° = cos 20° cos (3π/5) = sin (π/5)

Given a triangle with angle 45° opposite 8 ft, side 4√2 ft opposite angle β, and side a opposite angle θ, find teh remaining sides and angles.

sin B/4√2 = sin 45°/8 sin B = 1/2, so ∠B = 30° ∠A = 180 - (35 + 30) = 105° sin 105/a = sin 45/8 = 8√2 sin 105° ft or 10.9 ft

Figure 3 shows a 3-4-5 right triangle with the angle θ in between 4 and 5. Compute sin θ and θ. For θ, express the answer both in radians, rounded to two decimal places, and in degrees, rounded to one decimal place.

sin θ = 3/5 to isolate θ, take the inverse sine of both sides: sin⁻¹(sin θ) = sin⁻¹(3/5) θ = sin⁻¹(3/5) = 0.64 radians, 36.9°

Show that the area of a triangle where height is h, base is b, the left acute angle is θ, and the adjacent angle is a is given by A = (1/2)ab sin θ

sin θ = h/a, so h = a sin θ A = (1/2)bh so A = (1/2)ba sin θ

What is an example of the right-triangle identity sin² θ + cos² θ = 1?

sin² (10°) + cos² (10°) = 1 sin² (π/5) + cos² (π/5) =1

Suppose that B is an acute angle and cos B = 2/5. Find sin B and tan B

sin² B + cos² B = 1 sin² B + (2/5)² = 1 sin B = √21/5 tan B = (√21/5)/(2/5) = √21/2

A ladder that is leaning against the side of a building forms an angle of 50° with the ground. If the foot of the ladder is 12 ft from the base of the building, how far up the right side of the building does the ladder reach?

tan 50° = y/12 y = 12 tan 50° y = 14 ft

angle of elevation

the angle between the line of sight and the horizontal

angle of depression

the angle for an object below the horizontal

What does the sine of an angle equal if two angles are complementary?

the cosine of the other angle

Describe sin, cos, and tan for two similar right triangles, where one is exactly 10x larger

the trigonometric functions have the same ratio but are 10x larger for similar triangles

Compute the length x in a triangle where the angle 120° has a side x opposite it and is bordered by sides 7 cm and 8 cm.

x² = 7² + 8² - 2(7)(8) cos 120° x = √169 = 13 cm

Given a triangle with 10.2 cm opposite a 75° angle, a side 62° opposite an ungiven side, and a side θ opposite a side y, find the length y.

θ = 180 - (75 + 62) = 43° y/sin 43° = 10.2/sin 75° y = 10.2 sin 43°/sin 75° y ≈ 7.2 in