Lesson 2: Evaluating All Six Trig Functions
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Find all six trig function values given one value and the quadrant
- Use the Pythagorean identity to find a second trig value
- Apply reciprocal relationships to find the remaining four values
- Work with exact values (no decimals) throughout the process
Strategy: From One Value to All Six
Given one trig value and the quadrant, you can find all six trig functions using these steps:
- Use the Pythagorean identity to find a second value (usually sin and cos)
- Determine the correct sign from the quadrant
- Use quotient identities (tan = sin/cos, cot = cos/sin) for tangent and cotangent
- Use reciprocal identities for csc, sec, cot
Worked Examples
Example 1: Given sin, Find All Six
Given sin θ = 3/5 and θ is in QII. Find all six trig functions.
Solution:
Step 1: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5
Step 2: In QII, cosine is negative: cos θ = −4/5
Step 3: tan θ = sin/cos = (3/5)/(−4/5) = −3/4
Step 4: csc θ = 1/sin = 5/3, sec θ = 1/cos = −5/4, cot θ = 1/tan = −4/3
Example 2: Given tan, Find All Six
Given tan θ = −5/12 and θ is in QIV. Find sin θ and cos θ.
Solution:
Step 1: Since tan = opp/adj, let opp = 5, adj = 12. Then hyp = √(25+144) = 13.
Step 2: In QIV: sin is negative, cos is positive.
sin θ = −5/13, cos θ = 12/13
csc θ = −13/5, sec θ = 13/12, cot θ = −12/5
Example 3: Given sec, Find sin
Given sec θ = −3 and θ is in QIII. Find sin θ.
Solution:
Step 1: cos θ = 1/sec θ = −1/3
Step 2: sin²θ = 1 − cos²θ = 1 − 1/9 = 8/9
Step 3: sin θ = ±√(8/9) = ±2√2/3. In QIII, sin is negative: sin θ = −2√2/3
The Six Functions as a Family
The six trig functions are grouped into three pairs of reciprocals:
sin and csc: sin θ · csc θ = 1
cos and sec: cos θ · sec θ = 1
tan and cot: tan θ · cot θ = 1
And two quotient relationships:
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Common Mistakes to Avoid
- Forgetting to consider the sign based on the quadrant after taking a square root.
- Confusing reciprocal with negative: csc θ = 1/sin θ, NOT −sin θ.
- Using the wrong identity: always start with the Pythagorean identity sin² + cos² = 1.
Check Your Understanding
1. Given cos θ = −5/13 and θ in QIII, find sin θ.
2. Given tan θ = 7/24 and θ in QI, find sec θ.
3. If csc θ = −2 and cos θ > 0, find tan θ.
Key Takeaways
- From one trig value + quadrant, you can find all six values.
- Use sin² + cos² = 1 as the primary tool.
- Always apply the correct sign based on the quadrant.
- Reciprocal and quotient identities fill in the remaining values.