Basic Trig Identities — ACT Math Guide

Q: What You Need to Know

• **Pythagorean Identity**: sin²θ + cos²θ = 1 (the foundation of all trig identities) • **Quotient Identities**: tan θ = sin θ/cos θ and cot θ = cos θ/sin θ • **Reciprocal Identities**: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ • **Co-function Identities**: sin θ = cos(90° - θ) and cos θ = s

Basic trig identities ACT questions appear 2-4 times on every test, making them essential for your ACT math score. These fundamental relationships between sine, cosine, and tangent help you solve complex trigonometry problems quickly. The ACT math section gives you 60 questions in 60 minutes, so knowing these identities by heart saves precious time. With the right approach, these questions become some of the most reliable points you can earn.

What You Need to Know

Pythagorean Identity: sin²θ + cos²θ = 1 (the foundation of all trig identities)

Quotient Identities: tan θ = sin θ/cos θ and cot θ = cos θ/sin θ

Reciprocal Identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Co-function Identities: sin θ = cos(90° - θ) and cos θ = sin(90° - θ)

Even/Odd Identities: cos(-θ) = cos θ and sin(-θ) = -sin θ

📐 KEY FORMULA: sin²θ + cos²θ = 1

⏱️ ACT TIME TIP: Memorize the Pythagorean identity variations: 1 - sin²θ = cos²θ and 1 - cos²θ = sin²θ

How to Solve Basic Trig Identities ACT Questions

Example Question 1 — Easy/Medium Difficulty

If sin θ = 3/5 and θ is in the first quadrant, what is cos θ?

A) 4/5

B) -4/5

C) 3/4

D) -3/4

E) 5/4

Solution:

Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1

Step 2: Substitute sin θ = 3/5: (3/5)² + cos²θ = 1

Step 3: Solve for cos²θ: 9/25 + cos²θ = 1, so cos²θ = 16/25

Step 4: Since θ is in the first quadrant, cos θ is positive: cos θ = 4/5

✅Answer: A — In the first quadrant, both sine and cosine are positive.

Example Question 2 — Hard Difficulty

If tan θ = -5/12 and sin θ > 0, which of the following equals sec θ?

A) 13/12

B) -13/12

C) 12/13

D) -12/13

E) 5/13

Solution:

Step 1: Since tan θ < 0 and sin θ > 0, θ is in the second quadrant

Step 2: Use the identity 1 + tan²θ = sec²θ: 1 + (-5/12)² = sec²θ

Step 3: Calculate: 1 + 25/144 = 169/144, so sec²θ = 169/144

Step 4: Therefore sec θ = ±13/12, but in the second quadrant, cosine (and secant) is negative

✅Answer: B — Remember that secant has the same sign as cosine in each quadrant.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting quadrant signs when finding trig values

✅Fix: Always identify the quadrant first, then determine which functions are positive or negative

❌Mistake: Mixing up reciprocal relationships (confusing csc with sec)

✅Fix: Remember "co-sec-ant" goes with sine, "sec-ant" goes with cosine

❌Mistake: Not simplifying radical expressions in final answers

✅Fix: ACT answer choices are always in simplest form, so simplify your work

❌Mistake: Using degrees instead of the Pythagorean identity for unknown angles

✅Fix: When you don't know the specific angle, use algebraic identities instead of calculator values

Practice Question — Try It Yourself

If cos θ = -2/3 and θ is in the third quadrant, what is sin θ?

A) √5/3

B) -√5/3

C) 2√5/3

D) -2√5/3

E) √13/3

Show Answer

Answer: B — Use sin²θ + cos²θ = 1 to find sin²θ = 1 - 4/9 = 5/9, so sin θ = ±√5/3. In the third quadrant, sine is negative.

Key Takeaways for the ACT

Master the Pythagorean identity and its two rearranged forms — they solve most ACT math basic trig identities problems

Remember that your calculator is allowed throughout the entire ACT Math section, but identities are faster than computing

Quadrant signs matter: memorize ASTC (All Students Take Calculus) for positive functions in each quadrant

ACT trig questions often give you one function value and ask for another — use identities to connect them

The ACT uses exact values in answer choices, not decimal approximations