um take home test...i have no idea what im doing
always sometimes never
1. sin2x=2sinx
2. cos (pie/3+x)=-sinx
3. if tan x=3 and tan y=2 then tan (x+y)=5
4. when you use the half angle formulas you choose whether to use a + or - by noting in which quadrant the angle lies
5. using a graphing calc to solve a system of trigonometric equations yields the exact results
and the list goes on so if anyone can help with this stuff let me know
Sure, I can help you with this! Let's go through each of the statements and determine if they are always true, sometimes true, or never true.
1. sin2x = 2sinx
To determine if this statement is always true, sometimes true, or never true, we can simplify both sides of the equation and see if they are equal.
Starting with the left side, we can use the double-angle identity for sine: sin2x = 2sinxcosx. Therefore, the statement becomes: 2sinxcosx = 2sinx.
Now, we can divide both sides of the equation by 2sinx: cosx = 1.
Since the value of cosx is not always 1, the statement sin2x = 2sinx is never true.
2. cos(pi/3 + x) = -sinx
To determine the validity of this statement, we can use the trigonometric identity for the cosine of a sum of angles: cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
Using this identity in the given statement, we have:
cos(pi/3 + x) = cos(pi/3)cos(x) - sin(pi/3)sin(x).
Since cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2, the statement becomes:
cos(pi/3)cos(x) - sin(pi/3)sin(x) = -sinx.
Substituting the values, we get (1/2)cos(x) - (sqrt(3)/2)sin(x) = -sinx.
Simplifying further, (1/2)cos(x) + (sqrt(3)/2)sin(x) = 0.
This is equivalent to sin(60 degrees + x) = 0 or sin(pi/3 + x) = 0.
Since sin(0) = 0, we can conclude that the statement cos(pi/3 + x) = -sinx is sometimes true.
3. if tan x = 3 and tan y = 2, then tan(x + y) = 5
To determine the validity of this statement, we can use the trigonometric identity for the tangent of a sum of angles: tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b)).
Using this identity in the given statement, we have:
tan(x + y) = (tan(x) + tan(y))/(1 - tan(x)tan(y)).
Substituting the given values, we get tan(x + y) = (3 + 2)/(1 - 3*2) = 5/-5 = -1.
Since tan(-45 degrees) = -1, we can conclude that the statement tan(x + y) = 5 is sometimes true.
4. When you use the half-angle formulas, you choose whether to use a + or - by noting in which quadrant the angle lies.
This statement is always true. When using the half-angle formulas, you determine the sign based on the quadrant in which the angle lies. If the angle is in the first or second quadrant, you choose the positive sign (+). If the angle is in the third or fourth quadrant, you choose the negative sign (-). This rule ensures the correct sign for the half-angle formulas.
5. Using a graphing calculator to solve a system of trigonometric equations yields the exact results.
This statement is sometimes true. While a graphing calculator can effectively approximate the solutions to a system of trigonometric equations, it may not always provide exact results. Exact solutions usually involve algebraic manipulation and may not always be accurately represented on a calculator's graphing feature.
I hope this helps clarify the given statements! Let me know if you have any further questions.