Trigonometry — Right Triangles, Unit Circle, and Graphs

Trigonometry connects angles to lengths and repeating motion. Start with a right triangle, then extend the same ideas to the unit circle so they work for any angle. Keep your calculator mode visible: degrees and radians are different inputs.

Right-Triangle Ratios

Choose one acute angle, usually called θ. The hypotenuse is always the side opposite the right angle. The other two side names depend on θ: the opposite side is across from it, and the adjacent side touches it without being the hypotenuse.

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

SOH CAH TOA

Worked example: find a missing side

A right triangle has an angle of 35° and a hypotenuse of 12 cm. Find the side opposite the 35° angle.

sin 35° = opposite / hypotenuse
sin 35° = x / 12
12 sin 35° = x
x = 12(0.5736...)
x ≈ 6.88 cm

Round only at the end. The answer is shorter than the hypotenuse, which is a useful reasonableness check.

Solving a triangle

To solve a right triangle means finding every missing side and angle. Suppose the legs are 8 m and 15 m.

Find the hypotenuse:
c² = 8² + 15²
c² = 64 + 225
c² = 289
c = 17 m

Find an acute angle θ opposite the 8 m side:
tan θ = 8 / 15
θ = tan−1(8 / 15)
θ ≈ 28.1°

Find the other acute angle:
90° − 28.1° = 61.9°

The superscript −1 means an inverse trig function, not a reciprocal. Use tan−1 or arctan on the calculator.

Degrees and Radians

A full turn is 360° or 2π radians. A radian is defined by arc length: on a circle of radius r, an angle of 1 radian cuts off an arc of length r. Radians are the standard unit in algebra, calculus, and graphing formulas.

180° = π radians
degrees to radians:  angle in radians = angle in degrees × π / 180
radians to degrees:  angle in degrees = angle in radians × 180 / π

For example, convert 150° to radians:

150° × π / 180
= 150π / 180
= 5π / 6 radians

The Unit Circle and Exact Values

The unit circle has radius 1 and centre (0, 0). At angle θ, the point on the circle is (cos θ, sin θ). This gives signs as well as values: cosine is the horizontal coordinate and sine is the vertical coordinate. Tangent is sin θ / cos θ when cosine is not zero.

Angle	Radians	cos θ	sin θ	tan θ
0°	0	1	0	0
30°	π/6	√3/2	1/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	1/2	√3/2	√3
90°	π/2	0	1	undefined

Use the reference angle and quadrant signs for the rest of the circle. For example, 150° has reference angle 30° and lies in Quadrant II, where sine is positive and cosine is negative:

sin 150° = 1/2
cos 150° = −√3/2
tan 150° = (1/2) / (−√3/2)
tan 150° = −√3/3

Basic Identities

An identity is true for every angle where both sides are defined. These three are the core set to recognise and use.

sin² θ + cos² θ = 1
tan θ = sin θ / cos θ
1 + tan² θ = sec² θ

For a quick check, if sin θ = 3/5 and θ is acute:

sin² θ + cos² θ = 1
(3/5)² + cos² θ = 1
9/25 + cos² θ = 25/25
cos² θ = 16/25
cos θ = 4/5

The positive square root is correct because cosine is positive for an acute angle. For an angle in another quadrant, check the sign separately.

Sine and Cosine Graphs

The graph of y = sin x repeats every 2π radians. It starts at (0, 0), reaches 1 at π/2, returns to 0 at π, reaches −1 at 3π/2, and returns to 0 at 2π. The graph of y = cos x has the same shape shifted left by π/2: it starts at (0, 1).

y = A sin(B(x − C)) + D

amplitude = |A|
period = 2π / |B|  (when x is in radians)
horizontal shift = C
midline = y = D

Example: analyse y = 3 cos(2x) − 1.

Amplitude = |3| = 3
Period = 2π / 2 = π
Midline: y = −1
Maximum = −1 + 3 = 2
Minimum = −1 − 3 = −4

Cosine begins at a maximum, so one cycle has key points (0, 2), (π/4, −1), (π/2, −4), (3π/4, −1), and (π, 2).

Applied Example: Measuring Height

From level ground, a student stands 32 m from a building. The angle of elevation from the student’s eye to the roof is 41°. The student’s eye is 1.6 m above the ground. Find the building height.

tan 41° = vertical rise / horizontal distance
tan 41° = x / 32
x = 32 tan 41°
x ≈ 32(0.8693...)
x ≈ 27.82 m

building height = rise + eye height
building height ≈ 27.82 + 1.6
building height ≈ 29.42 m

To the nearest tenth, the building is 29.4 m tall. Adding eye height matters because the triangle starts at eye level, not at the ground.

Common Mistakes

• Using the wrong calculator mode: enter 41 in degree mode for 41°; use radian mode for expressions involving π.
• Misidentifying the hypotenuse: it is opposite the 90° angle, regardless of which acute angle you chose.
• Reversing opposite and adjacent: rename them relative to the marked angle every time.
• Forgetting a context length: add or subtract eye height, platform height, or depth only when the diagram and wording require it.
• Writing tan 90° as zero: it is undefined because it would require division by cos 90° = 0.

Practice Set

1. In a right triangle, θ = 28° and the adjacent side is 9 cm. Find the hypotenuse to the nearest tenth.
2. Convert 225° to radians in simplest form.
3. Find the exact values of sin 135° and cos 135°.
4. If cos θ = 5/13 and θ is acute, find sin θ.
5. For y = −2 sin(3x) + 4, state the amplitude, period, and midline.

Answers

1. cos 28° = 9/h, so h = 9/cos 28° ≈ 10.2 cm.
2. 225° × π/180 = 5π/4.
3. Reference angle 45° in Quadrant II: sin 135° = √2/2 and cos 135° = −√2/2.
4. sin² θ = 1 − (5/13)² = 144/169, so sin θ = 12/13.
5. Amplitude 2; period 2π/3; midline y = 4.