Question: State an equivalent Trigonometric Ratio for each of the following
a) cos π/16
b) sin 7π/9
c) tan 5π/4
d) sin 11π/6
My (attempted) answers are:
a) -cos π/16
b) sin 2π/9
c) tan π/4
d) -sin π/6
Are any of my attempted answers correct?
1. No, cos π/16 ≠ -cosπ/16, what you probably meant is cos π/16 = cos (-π/16)
2. yes
3. yes
4. yes
Let's go through each trigonometric ratio and see if your attempted answers are correct.
a) cos π/16:
The equivalent trigonometric ratio for cos π/16 is sin π/2 - π/16 since cos θ = sin (π/2 - θ). Therefore, your attempted answer of -cos π/16 is incorrect.
b) sin 7π/9:
The equivalent trigonometric ratio for sin 7π/9 is cos (π/2 - 7π/9) since sin θ = cos (π/2 - θ). Therefore, your attempted answer of sin 2π/9 is incorrect.
c) tan 5π/4:
The equivalent trigonometric ratio for tan 5π/4 is -1 since tan θ = sin θ / cos θ, and sin 5π/4 = -√2/2 and cos 5π/4 = -√2/2. Therefore, your attempted answer of tan π/4 is incorrect.
d) sin 11π/6:
The equivalent trigonometric ratio for sin 11π/6 is -1/2 since sin θ = sin (θ - 2π) and sin (11π/6 - 2π) = sin (5π/6) = 1/2. Therefore, your attempted answer of -sin π/6 is incorrect.
To summarize:
a) cos π/16: The correct answer is sin π/2 - π/16.
b) sin 7π/9: The correct answer is cos (π/2 - 7π/9).
c) tan 5π/4: The correct answer is -1.
d) sin 11π/6: The correct answer is -1/2.
Let's go through each of the trigonometric ratios and check if your answers are correct.
a) The equivalent ratio for cos(pi/16) can be found by using the cofunction identity for cosines: cos(pi/2 - x) = sin(x). In this case, we have cos(pi/16) = sin(pi/2 - pi/16) = sin(15pi/32). So, your answer of -cos(pi/16) is incorrect.
b) The equivalent ratio for sin(7pi/9) can be found by using the periodicity of the sine function: sin(x) = sin(x + 2kpi), where k is an integer. So, we can find an angle between 0 and 2pi that is congruent to 7pi/9. In this case, 7pi/9 is equivalent to 5pi/9 (by subtracting 2pi). Therefore, sin(7pi/9) = sin(5pi/9). So, your answer of sin(2pi/9) is incorrect.
c) The equivalent ratio for tan(5pi/4) can be found by using the periodicity of the tangent function: tan(x) = tan(x + kpi), where k is an integer. In this case, 5pi/4 is equivalent to 5pi/4 + 2pi = 13pi/4, which falls into the second quadrant. In the second quadrant, the value of tan(x) is negative. So, the equivalent ratio for tan(5pi/4) is -tan(pi/4). Therefore, your answer of tan(pi/4) is incorrect.
d) The equivalent ratio for sin(11pi/6) can also be found by using the periodicity of the sine function. 11pi/6 is equivalent to 11pi/6 - 2pi = 5pi/6, which is congruent to pi/6. In the first quadrant, the value of sine is positive. So, the equivalent ratio for sin(11pi/6) is sin(pi/6). Therefore, your answer of -sin(pi/6) is incorrect.
To summarize:
a) cos(pi/16) is equivalent to sin(15pi/32)
b) sin(7pi/9) is equivalent to sin(5pi/9)
c) tan(5pi/4) is equivalent to -tan(pi/4)
d) sin(11pi/6) is equivalent to sin(pi/6)
Make sure to double-check your answers for trigonometric ratios using identities and periodicity rules when needed.