Pre-Calculus — Semester B
Free Practice · 10 Questions · 20 min
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Question 1 of 10
TEKS 4A-4GEasy Diagram
√31230°60°

In the 30-60-90 triangle shown with hypotenuse 2, what is sin(30°)?

A√3
B2
C√3/2 — the side adjacent to 30° over the hypotenuse
D1/2 (opposite/hypotenuse)
Explanation
30-60-90 special triangle: sides in ratio 1 : √3 : 2 (opposite 30°, 60°, 90°). For the 30° angle: opposite = 1, hypotenuse = 2. sin(30°) = 1/2. The value √3/2 is sin(60°) or cos(30°).
Question 2 of 10
TEKS 4H-4KEasy

Express tan θ in terms of sin θ and cos θ.

Atan θ = sin θ · cos θ
Btan θ = sin θ / cos θ
Ctan θ = cos θ / sin θ
Dtan θ = sin θ + cos θ
Explanation
Definition: tan θ = sin θ / cos θ. The reversed ratio cos θ / sin θ is the reciprocal: cot θ. Visualize: tan = opposite/adjacent = (opposite/hyp) / (adjacent/hyp) = sin/cos.
Question 3 of 10
TEKS 5A-5CEasy Diagram
5n=18n=211n=314n=417n=5Each term adds 3 → common difference d = 3

The bars show an arithmetic sequence: 5, 8, 11, 14, 17. Find the formula for the nth term aₙ.

Aaₙ = 5n
Baₙ = 5 + 3(n − 1) = 3n + 2
Caₙ = 5 + 3n, applying d·n without the (n − 1) shift
Daₙ = 5 · 3ⁿ
Explanation
Arithmetic sequence formula: aₙ = a₁ + d(n − 1), where a₁ is the first term and d is the common difference. Here a₁ = 5, d = 3. So aₙ = 5 + 3(n − 1) = 3n + 2. Verify: a₁ = 3(1)+2 = 5 ✓; a₂ = 3(2)+2 = 8 ✓; a₃ = 11 ✓.
Question 4 of 10
TEKS 4A-4GEasy Diagram
11√245°45°

In the 45-45-90 triangle shown with legs of length 1, what is sin(45°)?

A1
B1/2
C√2/2 (= 1/√2)
D√2
Explanation
45-45-90 special triangle: isosceles right triangle with sides in ratio 1 : 1 : √2. sin(45°) = opposite/hypotenuse = 1/√2 = √2/2. cos(45°) = √2/2 (same — because the triangle is symmetric). tan(45°) = 1.
Question 5 of 10
TEKS 4A-4GEasy

Convert π/3 radians to degrees.

A60°
B30°
C45°
D90°
Explanation
π radians = 180°. So π/3 = 180°/3 = 60°. Use the conversion factor (180/π) to go from radians to degrees: (π/3) · (180/π) = 60.
Question 6 of 10
TEKS 4A-4GEasy

Convert 90° to radians.

Aπ/2
Bπ
Cπ/4
D2π
Explanation
Conversion: 180° = π radians. So 90° = π/2. Other common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 180° = π, 360° = 2π.
Question 7 of 10
TEKS 4A-4GEasy Diagram
b = 4a = 3c = 5θ

In the 3-4-5 right triangle shown, the angle θ is at the bottom-right vertex (adjacent to side b = 4 and opposite side a = 3). What is sin θ?

Asin θ = 4/5 (adjacent/hypotenuse)
Bsin θ = 3/5 (opposite/hypotenuse)
Csin θ = 3/4 (opposite/adjacent)
Dsin θ = 5/3
Explanation
SOH-CAH-TOA: sin θ = opposite / hypotenuse. From angle θ's perspective: opposite side = 3 (vertical leg), hypotenuse = 5. So sin θ = 3/5. The ratio 4/5 is cos θ; 3/4 is tan θ; 5/3 inverts the ratio.
Question 8 of 10
TEKS 4A-4GEasy Diagram
-2-112-112xy(1, 0)(0, 1)(-1, 0)(0, -1)IIIIIIIV

On the unit circle shown, what are the coordinates of the point corresponding to θ = π/2 (90°)?

A(0, 1)
B(1, 0)
C(0, −1)
D(−1, 0)
Explanation
Unit circle definition: at angle θ, the point on the circle is (cos θ, sin θ). At θ = π/2: cos(π/2) = 0, sin(π/2) = 1. So point = (0, 1) — the top of the circle. The point (1, 0) is θ = 0; the point (−1, 0) is θ = π.
Question 9 of 10
TEKS 4H-4KEasy

Which is the PYTHAGOREAN IDENTITY?

Atan θ = sin θ · cos θ
Bsin²θ + cos²θ = 1
Csin(2θ) = 2sin θ
Dsin θ + cos θ = 1
Explanation
Pythagorean identity: sin²θ + cos²θ = 1 — comes directly from the unit circle (x² + y² = 1 with x = cos θ, y = sin θ). Derived: dividing through by cos²θ gives tan²θ + 1 = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.
Question 10 of 10
TEKS 4A-4GEasy Diagram
-4-3-2-11234-2-112xyπ2π-π

The graph above shows y = sin(x). What is the PERIOD of this function?

Aπ
B2π
Cπ/2
D4π
Explanation
Period of sin(x): the smallest positive T such that sin(x + T) = sin(x) for all x. From the graph: one complete wave cycle spans 2π (e.g., from 0 to 2π, the function goes 0 → 1 → 0 → −1 → 0). For sin(Bx): period = 2π/|B|.

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