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CHAPTER 4
TRIGONOMETRIC FUNCTIONS

4.1 Angles and Their Measures

* vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is equal to the central angle itself.

* Degree is a unit of angular measure equal to 1/180th of a straight angle.

* Minutes is a unit of measurement oftime or of angle. The minute is a unit of time equal to 1/60 of an hour or 60 seconds.

* Seconds is the base unit of time in the International System of Units (SI) and is also a unit of time in other systems of measurement.

* Radian a central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

* Degree Radian Conversion- To convertradians to degrees, multiply by 180°⁄ π radians. To convert degrees to radians, multiply by π radians⁄ 180°.

* Arc Length Formula (Radian Measure) is a central angle in a circle of radius r, and if θ is measured in radians, then the length s of the intercepted arc is given by s = rθ.

* Arc Length Formula (Degree Measure) is a central angle in a circle of radius r, and if θ is measuredin degrees, then the length s of the intercepted arc is given by s = πrθ ⁄ 180.

* Nautical mile is the length of 1 minute of arc along Earth’s equator.

4.2 trigonometric functions of acute angles


* Similar Figure they have the same shape even thought they may have different sizes.

* Standard Position of an Angle in the xy-plane, with the vertex at theorigin, one ray along the positive x-axis, and the other ray extending into the first quadrant.

Trigonometric Functions
Let θ be an acute angle in the right ΔABC. Then

Sine θ = opp/hyp cosecant θ = hyp⁄opp

Cosine θ = adj/hyp secant θ = hyp/adj

Tangent θ = opp/adj cotangent θ = adj/opp

* Solving a Triangle is using some of theparts of a triangle to solve for all the others.

4.3 Trigonometric Function of Any Angle

* Initial of An Angle the beginning position of the ray.

* Vertex of An Angle endpoint of the ray.

* Terminal Side final position of the ray.

* Measure of An Angle is a number that describes the amount of rotation from the initial side to the terminal side of the angle.

*Positive Angles is angles generated by counterclockwise rotations.

* Negative Angles is angles generated by clockwise rotation.

* Coterminal Angles two angles in this expanded angle-measurement system can have the same initial side and the same terminal side, yet have different measures.

* Reference Triangle is a 45° −45° −90° triangle.

* Quadrantal Angles angles whoseterminal sides lie along one of the coordinate axes.

4.8 Solving Problems with Trigonometry


* Angle of Elevation is the angle through which the eye moves the eye move up from horizontal to look at something, above.

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EXERCISES

61. a guy wire fram the top of the transmission tower at WJBC forms a 75º angle with the ground at a 55- foot distance from the base of the tower. how tall is the tower?

62. Kirsten places her surveyor’s telescope on the top of a tripod 5 feet above the ground. She measures an 8° elevation above the horizontal to the top of a tree that is 120 feet away. How tall is the tree?



63. For locations between 20° and 60° north latitudea solar collector panel should be mounted so that its angle with the horizontal is 20 greater than the local latitude. Consequently, the solar panel mounted on the roof of Solar Energy, Inc., in Atlanta (latitude 34°) forms a 54° angle with the horizontal. The bottom edge of the 12-ft-long panel is resisting on the roof, and the high edge is 5 ft. above the roof. What is the total area of this...