13-1 Use Trigonometry with Right Triangles Name:______

Objective: To use trigonometric functions to find lengths. Algebra 2

* The six trigonometric functions: sine (sin), cosine(cos), tangent(tan),
cosecant (csc), secant (sec), cotangent (cot)
*Must know and use:
  • Definitions of trig functions: (SOH CAH TOA)
  • Pythagorean Theorem:

*RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS
Let  be an acute angle of a right triangle.
The six trigonometric functions of are defined as follows:
sin csc
cos sec
tan cot
The abbreviations opp, adj, and hyp are often used to represent the side lengths of the right triangle. Note that the ratios in the second column are reciprocals of the ratios in the first column:
csc sec cot

Ex. 1: Evaluate the six trigonometric functions of the angle 0.

Ex 2: Find the other trig functions when sin .

(make sure to draw a triangle to solve trig problems!)

*Memorize special right triangles--- They are hugely important in this chapter!

( ) ( )
*Trigonometric values for special angles

/ sin / cos / tan



Ex. 3: (Use a calculator to solve a right triangle) You Try: Solve ABC.

Solve ABC.

Ex. 4: (Use an angle of elevation)

You are measuring the height of your school building. You stand 25 feet from the base of the school. The angle of elevation from a point on the ground to the top of the school is 62°. Estimate the height of the school to the nearest foot.

You Try: A kite makes an angle of 59° with the ground. If the string on the kite is 40 feet, how far above the ground is the kite itself? Round to the nearest foot.

Ex. 5: You measure the angle of elevation from the ground to the top of a building as 22°. When you move 3 meters closer to the building, the angle of elevation is 25°. How high is the building? Show

a labeled diagram and calculation below.

You Try: You measure the angle of elevation from the ground to the top of a building as 50°. When you move 50 meters away from the building, the angle of elevation is 30°. How high is the building?

13-2Define General Angles and Use Radian Measure

Objective: To use general angles that may be measured in radians.

*Angles in Standard Position
In a coordinate plane, an angle can be formed by fixing one ray,
called the initial side, and rotating the other ray,
called the terminal side, about the vertex.
An angle is in standard position if its vertex is at origin and its initial side lies on the positive x-axis.
  • counter clockwise : (+) measure
  • clockwise : (-) measure

Ex 1: Draw an angle with the given measure in standard position.

a.405° / b. -65°

Ex 2: (Find coterminal angles)

Find one spositive angle and one negative angle that are coterminal with 210°.

---There are many such angles, depending on what multiple of 360° is added or subtracted.

210° + 360° = ______210°  360° = ______

You Try: Draw an angle with the given measure in standard position. Then find one smallest

positive coterminal angle and one smallest negative coterminal angle.

1.485° 2. 75°

* 1 radian angle:an angle whose arc length equals to the radius (about 57.3°)
* Any angle measure not labeled degree ( °) is radian!
Radian is usually given in terms of .
* Unit Circle : a circle with radius of 1
* Converting between Degrees and Radians
  • Degrees to Radians : Multiply by
  • Radians to Degrees : Multiply by

Ex 3: Convert. (a) 315° to radians (b)radians to degrees.

You Try: Convert. (a) 200° to radians (b)radians to degrees.

*Sector : A region of a circle that is bound by two radii and ar arc of the circle
*Arc length and area of a sector
The arc length s and area A of a sector with radius r and
central angle (measured in radians) are as follows.
Arc length: s = r
Area :

Ex 4 :Find the arc length and area of a sector with a radius of 15 inches and a central angle of 60°.

You Try : Find the arc length and area of the sector with given radius and angle.

r = 5 ft,  = 75o

Ex 5 :Draw an angle with the given measure in standard position. Then find one smallest

positive coterminal angle and one smallest negative coterminal angle.

(If given in radians, answer in radians, too!)

(a) (b)

Ex 6 : Evaluate secwithout using a calculator.

You Try:Evaluate cot using a calculator.

<Discussion> Which mode do you need to use? Radian or degree mode?

  1. Evaluate sin 13°.
  1. Evaluate sin
  1. Find the area of the sector if r = 5 ft,  =

Algebra 2 Ch. 13ANotes_Page 1