In the second quadrant, sine is positive while cosine and tangent are negative. Using sinx=53 and the Pythagorean identity sin2x+cos2x=1, we find cosx=−54, then compute the remaining four functions from these two values.
Why the quadrant matters
Trigonometric functions have fixed sign patterns in each quadrant. In the second quadrant (90∘<x<180∘):
- sinx is positive
- cosx is negative
- tanx is negative (positive divided by negative)
- cscx, secx, cotx follow the signs of their reciprocals
Since we're given sinx=53 and told x is in the second quadrant, we know immediately that cosx and tanx must be negative. This sign information is not optional — it's essential for getting the correct answer.
A common mistake is to find cosx=±54 from the identity and pick the positive sign out of habit. The quadrant tells you which sign to choose — always check it.
Step-by-step solution
1. Find cosx using the Pythagorean identity
The fundamental identity is:
sin2x+cos2x=1
Substitute sinx=53:
(53)2+cos2x=1
259+cos2x=1
cos2x=1−259=2516
Taking the square root gives cosx=±54. Since x is in the second quadrant, cosx is negative:
cosx=−54
2. Find tanx
tanx=cosxsinx=−4/53/5=53×(−45)=−43
3. Find cscx (reciprocal of sinx)
cscx=sinx1=3/51=35
Since sinx is positive in QII, cscx is also positive.
4. Find secx (reciprocal of cosx)
secx=cosx1=−4/51=−45
5. Find cotx (reciprocal of tanx)
cotx=tanx1=−3/41=−34
You can also find cotx directly as sinxcosx=3/5−4/5=−34. This is often faster than computing tanx first and then taking its reciprocal.
Final result
Here are all six trigonometric functions for the given x:
| Function | Value |
|---|
| sinx | 53 |
| cosx | −54 |
| tanx | −43 |
| cscx | 35 |
| secx | −45 |
| cotx | −34 |
✓Final answer
The other five trigonometric functions are cosx=−54, tanx=−43, cscx=35, secx=−45, and cotx=−34.