Chapter 6

1. Properties of Sin Function : - The X-intercepts are located πn, where n is an integer - The Y-intercept is 0 - The maximum values are y=1 and occur when x = π/2 + 2πn , when n is an integer -The minimum values are y = -1 and occur when x = 3π/2+2πn, where n is an integer

2. Properties of the Cosine function: - The domain is the set of all real numbers - The range consists of all real numbers from -1 to 1 - The cosine function is an even function, as the symmetry of the graph with respect to the Y axis indicates -The cosine function is periodic with a period of 2π - The X intercepts are -3π/2 , -π/2 , π/2 , 3π/2 , 5π/2 - The maximum of -1 occurs at x= ..-π,π,3π,5π

3. Properties of Tangent Function: - Domain : all real x - Range : ( -infinity , +infinity ) - Period : π - Vertical asymptotes: X= π/2+nπ(N=Z

4. - Determine the Domain of the Trigonometric Functions: • The domain of the sine function is the set of all numbers • The domain of the cosine function is the set of all numbers • The domain of the tangent function is the set of all real numbers except odd integer multiplies of π/2 (90®) • The domain of the secant function is the set of all real numbers except odd integer multiplies of π/2 (90) • The domain of the cotangent function is the set of all real numbers except integer multiplies of π (180) • The domain of the cosecant function is the set of all real numbers except integer multiplies of π (180)

5. - The Values of the Trigonometric Functions Using Fundamental Identities: • Sin2Ɵ + Cos2Ɵ= 1 • Tan2 Ɵ+1= sec2 Ɵ • Cot2 Ɵ+1= Csc2 Ɵ - Example: - Q3) Tan 20- sin 20/cos 20 - ANS) Tan 20 – Tan 20= 0

6. Finding the values of the trigonometric functions of Ɵwhen the value of one function is known and the quadrant of Ɵis known: Option 1: using a circle of radius r Step 1: draw a circle centered at the origin showing the location of the angle Ɵ and the point P= (x,y) that corresponds to Ɵ and r= √x2+y2 Step 2: assign a value to two of the three variables x,y,r based on the value of the given trigonometric function and the location of P Step 3: use the fact that P lies on the circle x2+y2= r2 to find the value of the missing variable. Step 4: apply the theorem we mentioned earlier to find the values of the remaining of the trigonometric functions

7. 6.3

8. 6.4

9. 6.5

10. - There are other trigonometric functions such as π/4=45® or π/6= 30® or π/3= 60® and they have fix values:

11. Sin t= y cos t= x tan t= y/x Csc t= 1/y sec t= 1/x cot t= x/y Example: Find the values of the six trigonometric Sin t= √3/2 cos t= -1/2 tan t= (√3/2)/(-1/2)= -√3 Csc t= 2√3/3 sec t= -2 cot t= -√3/3

12. - An angle Ɵ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis.

13. - The two commonly used measures for angles are degrees and radians - Arc Length: For a circle of radius (r), a central angle of Ɵ radians subtends an arc whose length is (s) is s= rƟ

14. 6.1

15. 6.2