7-3: COMPUTING THE VALUES OF TRIG FUNCTIONS
In a right triangle, if one of the acute angles = 45(, then so does the other; and the triangle is isosceles and could have legs = 1, hypotenuse = [pic]. In a right triangle, if one of the acute angles is 30(, then the other is 60(; such a triangle could have a hypotenuse of 2 and legs of 1 and [pic].
Find the exact value of the six trigonometric functions of 45(, 30(, and 60(:
| |Sine |Cosine |Tangent |Cotangent |Cosecant |Secant |
|45(=(/4 | | | | | | |
|30(=(/6 | | | | | | |
|60(=(/3 | | | | | | |
Find the exact value of each expression if ( = 60(; do not use a calculator:
1. [pic] 2. 3 csc ( 3. [pic]
Find the exact value of each expression; do not use a calculator:
4. 4 sin 45( + 2 cos 30( 5. 5 tan 30( . sin 60( 6. 1 + sec2 45( - cos2 60(
Use a calculator to find the approximate value of each expression; round to 2 decimal places:
7. cos 42( 8. sec 38( 9. csc 72(
10. [pic] (use radian mode) 11. [pic] 12. tan 42.859(
Projectile Motion, fired at inclination ( and initial speed v0 (g ( 32.2 ft/sec2 ( 9.8 m/sec2: Horizontal distance: [pic] Height: [pic]
13. Find the range R and maximum height H of a projectile fired at an angle of 40( to the horizontal with an initial speed of 300 m/sec.
14 In a certain piston engine, the distance x (in meters) from the center of the drive shaft to the head of the piston, where ( is the angle between the crank and the path of the piston head by the formula below. Find x when ( = 35(. [pic]
7.4 TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
Let ( be any angle in standard position and let (a, b) denote any point except the origin on the terminal side of (. If [pic]denotes the distance from (0, 0) to (a, b), then the six trigonometric functions of ( are defined as the following ratios: [pic] [pic] [pic]
Quandrantal angles are angles whose terminal side lies on the x- or y-axis, such as 0(, 90(, 180(, 270, and 360(. Their trig function values will always be 0, (1, or undefined.
A point on the terminal side of an angle ( is given. Find the exact value of the six trigonometric functions of the angle (:
| |(a, b) |
|Sin ( < 0 |Sin ( < 0 |
|Cos ( < 0 |Cos ( > 0 |
|Tan ( > 0 |Tan ( < 0 |
3. sin ( > 0, cos < 0 4. tan ( < 0, sec > 0 5. csc < 0, cot > 0
Two angles in standard position are coterminal if they have the same terminal side. If ( is a non-acute angle, the acute angle formed by the terminal side of ( and the ( x-axis is called the reference angle for (. Find the reference angle of each angle and name its quadrant:
6. 300( 7. -490( 8. [pic] 9. [pic]
A general angle ( and its coterminal reference angle ( have the same values of their trig functions except for the sign, which depends the quadrant in which it lies. Find the exact value of each expression without a calculator:
|10 – 13. |cos (-420() |sec 630( |csc [pic] |sin [pic] |
|Quadrant | | | | |
|Reference Angle | | | | |
|Exact Value | | | | |
Find the exact value of each of the remaining trigonometric functions of (:
|14 – 16. | |Cot ( < 0 |Tan ( > 0 |
|Quadrant |IV | | |
|(a, b), r | | | |
|Sin ( | | | |
|Cos ( |[pic] | | |
|Tan ( | | | |
|Csc ( | |3 | |
|Sec ( | | | |
|Cot ( | | | |
