10.3 Trigonometry Essentials

Trigonometry on the USTET

Grade 11 General Mathematics and Precalculus introduce trigonometry as the study of relationships between angles and sides. On the USTET Mathematics subtest, expect right-triangle trig: naming opposite/adjacent/hypotenuse, evaluating sin/cos/tan (and sometimes reciprocal ratios), using exact values of special angles, and solving short elevation or depression word problems. Calculators are not assumed — exact radical answers and the SOH-CAH-TOA definitions matter more than decimal approximations.

Right-Triangle Setup

In a right triangle, the hypotenuse is the side opposite the right angle (always longest). Relative to an acute angle θ:

• Opposite — the side across from θ
• Adjacent — the side next to θ that is not the hypotenuse

Switching which acute angle you use swaps opposite and adjacent; the hypotenuse stays the same.

Primary and Reciprocal Ratios

Also useful: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.

Worked Example 1. In right △ABC with right angle at C, AC = 8 (adjacent to ∠A), BC = 15 (opposite ∠A), and AB = 17. Find sin A, cos A, and tan A.

Recognize the 8-15-17 triple. sin A = 15/17, cos A = 8/17, tan A = 15/8.

Worked Example 2. If sin θ = 3/5 and θ is acute, find cos θ and tan θ.

Opposite = 3, hypotenuse = 5 → adjacent = 4 (3-4-5 triangle). cos θ = 4/5, tan θ = 3/4. Using the identity: cos θ = √(1 − sin²θ) = √(1 − 9/25) = √(16/25) = 4/5.

Exact Values of Special Angles

Memorize this table — USTET options often list radicals, not decimals.

Pattern tip: sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2. Cosine reads the same list in reverse. Tangent = sin/cos.

These values match the 30°-60°-90° and 45°-45°-90° side ratios from the previous geometry section: for 30°, opposite/hypotenuse = x/(2x) = 1/2; for 60°, (x√3)/(2x) = √3/2; for 45°, x/(x√2) = 1/√2 = √2/2.

Worked Example 3. Evaluate sin 60° · cos 30° − cos 60° · sin 30°.

(√3/2)(√3/2) − (1/2)(1/2) = 3/4 − 1/4 = 1/2. (This expression is sin(60° − 30°) = sin 30°, a preview of the sine difference formula — useful if you recognize it, not required if you expand with the table.)

Worked Example 4. If cos θ = 1/2 and θ is acute, find θ and tan θ.

From the table, θ = 60°. Then tan 60° = √3.

The Pythagorean Identity

For any angle θ where the ratios are defined:

sin²θ + cos²θ = 1

Divide by cos²θ (when cos θ ≠ 0) to get 1 + tan²θ = sec²θ. Divide by sin²θ (when sin θ ≠ 0) to get 1 + cot²θ = csc²θ. The primary identity is the one you need most for USTET.

Worked Example 5. If cos θ = 5/13 and θ is acute, find sin θ.

sin θ = √(1 − 25/131) wait — 1 − (5/13)² = 1 − 25/169 = 144/169, so sin θ = 12/13. The 5-12-13 triple appears again.

Solving for a Missing Side

1. Sketch the right triangle; mark the given angle and the right angle.
2. Label the known side and the unknown side as opposite, adjacent, or hypotenuse relative to the given acute angle.
3. Choose the ratio that involves the known and the unknown.
4. Write the equation and solve. Rationalize denominators when options are rationalized.

Worked Example 6. From the foot of a vertical flagpole, the angle of elevation to the top is 45°. The distance from the foot to the observer is 12 m. How tall is the flagpole?

tan 45° = height / 12. Since tan 45° = 1, height = 12 m. (A 45° elevation with adjacent 12 forces an isosceles right triangle.)

Worked Example 7. A ladder leans against a wall and makes a 60° angle with the ground. The ladder is 10 m long. How high up the wall does it reach?

sin 60° = height / 10 → height = 10 × (√3/2) = 5√3 m.

Worked Example 8. In a right triangle, an acute angle measures 30° and the adjacent leg is 8. Find the opposite leg and the hypotenuse.

tan 30° = opp/8 = 1/√3 → opp = 8/√3 = (8√3)/3. Hypotenuse: cos 30° = 8/hyp = √3/2 → hyp = 8 × 2/√3 = 16/√3 = (16√3)/3. Or use 30°-60°-90°: adjacent to 30° is the long leg x√3 = 8, so x = 8/√3; hypotenuse 2x = 16/√3.

Angles of Elevation and Depression

• Angle of elevation: the angle upward from the horizontal to the line of sight.
• Angle of depression: the angle downward from the horizontal to the line of sight.

If person A looks up to person B at elevation α, then B looking down to A has depression α — the two angles are equal (alternate interior angles formed by a transversal across parallel horizontals). Draw the horizontal carefully; students often measure from the vertical by mistake.

Worked Example 9. From the top of a 30 m building, the angle of depression to a jeepney on the street is 30°. How far is the jeepney from the base of the building?

The depression angle equals the elevation angle from the jeepney to the top. tan 30° = 30 / distance → distance = 30 / (1/√3) = 30√3 m.

Worked Example 10. An observer 1.5 m tall stands 20 m from a tree. The angle of elevation to the top of the tree is 45°. Approximate the tree's height.

tan 45° = (tree height − 1.5) / 20 = 1 → tree height − 1.5 = 20 → tree height = 21.5 m. Always check whether the problem includes the observer's eye height.

Cofunction Relationships

In a right triangle, the two acute angles are complementary (sum to 90°). Therefore:

• sin θ = cos(90° − θ)
• cos θ = sin(90° − θ)
• tan θ = cot(90° − θ)

So sin 30° = cos 60°, and cos 45° = sin 45°. This is why the sine and cosine rows of the special-angle table are reverses of each other.

Common Traps

• Swapping opposite and adjacent for the angle in the question (label relative to that angle).
• Using tan when the hypotenuse is involved (choose sin or cos instead).
• Leaving tan 90° or sec 90° as a number — they are undefined.
• Forgetting to rationalize or match the exact form in the options (√2/2 vs 1/√2).
• Measuring elevation/depression from the vertical instead of the horizontal.
• Using sin = adjacent/hypotenuse (that is cosine).

Section Takeaway

SOH-CAH-TOA defines the three primary ratios; reciprocals flip them. Memorize the 0°–90° exact-value table and sin²θ + cos²θ = 1. To solve sides, pick the ratio that links the known side to the unknown. Elevation and depression are horizontal-referenced and equal along the same line of sight. Together with the geometry and measurement sections, this completes the USTET Mathematics geometry-and-trigonometry block.