I have the trigonometric equations:
y=3+2sec(pi/8)(x-3)
y=12+4cot(pi/6)(x-1)
y=3+7csc(pi/4)(x-1)
how do I make these into sine, cosine, and tangent equations so that I can work with them more easily?
Is the last parentheses part of the angle?
for example, do you mean
y=3+2sec[(pi/8)(x-3)] ?
If not, just substitute 1.0824 for sec pi/8 etc.
To convert the given trigonometric equations into equations involving sine, cosine, and tangent, we need to recall the definitions and relationships among the trigonometric functions.
Let's start with the equation involving secant (sec):
y = 3 + 2sec(pi/8)(x - 3)
Recall that secant is the reciprocal of cosine (cos):
sec(theta) = 1 / cos(theta)
We can replace sec(pi/8) with its reciprocal, which is 1 / cos(pi/8):
y = 3 + 2(1 / cos(pi/8))(x - 3)
Now we need to express this equation in terms of sine, cosine, and tangent. We can use the Pythagorean identity:
cos²(theta) + sin²(theta) = 1
Rearranging this equation, we get:
1 / cos²(theta) = 1 + sin²(theta) / cos²(theta)
Now, replacing cos(pi/8) with its equivalent in terms of sine using the above identity, we get:
y = 3 + 2(1 / (1 + sin²(pi/8) / cos²(pi/8)))(x - 3)
This equation involves both sine and cosine.
Moving on to the equation involving cotangent (cot):
y = 12 + 4cot(pi/6)(x - 1)
Recall that cotangent is the reciprocal of tangent (tan):
cot(theta) = 1 / tan(theta)
So, we can replace cot(pi/6) with its reciprocal, which is 1 / tan(pi/6):
y = 12 + 4(1 / tan(pi/6))(x - 1)
Now we need to express this equation in terms of sine and cosine. Recall the relationship between tangent and sine over cosine:
tan(theta) = sin(theta) / cos(theta)
We can replace tan(pi/6) with its equivalent in terms of sine and cosine:
y = 12 + 4(1 / (sin(pi/6) / cos(pi/6)))(x - 1)
Simplifying further:
y = 12 + 4(cos(pi/6) / sin(pi/6))(x - 1)
This equation involves both sine and cosine.
Lastly, let's consider the equation involving cosecant (csc):
y = 3 + 7csc(pi/4)(x - 1)
Recall that cosecant is the reciprocal of sine (sin):
csc(theta) = 1 / sin(theta)
We can replace csc(pi/4) with its reciprocal, which is 1 / sin(pi/4):
y = 3 + 7(1 / sin(pi/4))(x - 1)
Now we need to express this equation in terms of sine only. Recall the relationship between sine and cosine:
sin²(theta) + cos²(theta) = 1
Rearranging this equation, we get:
1 / sin²(theta) = 1 - cos²(theta) / sin²(theta)
Now, replacing sin(pi/4) with its equivalent in terms of cosine using the above identity, we get:
y = 3 + 7(1 / (1 - cos²(pi/4) / sin²(pi/4)))(x - 1)
This equation involves sine only.
By making these conversions, you can now work with the equations more easily by using the relationships and properties of sine, cosine, and tangent.