Choose the angle and label the sides
In a right triangle, the hypotenuse lies opposite the right angle. Relative to an acute angle θ, the opposite side lies across from θ and the adjacent side touches θ without being the hypotenuse. Define and tan θ = opposite/adjacent. Changing the chosen angle interchanges opposite and adjacent, but not hypotenuse. A labelled sketch is therefore part of the reasoning, even when the numerical calculation is short.
To find a missing side, select the ratio containing that side and a known side. If a ladder is 10 m long and makes 60° with horizontal ground, its vertical reach is, approximately 8.66 m. Its horizontal reach is 10 cos 60° = 5 m. To find a missing acute angle, use an inverse ratio: if opposite/hypotenuse = 0.6. Here sin⁻¹ means inverse sine, not 1/sin θ.
Use exact angles and correct calculator mode
From standard right triangles, and tan 45° = 1. Also and cos 60° = 1/2. Keeping these values exact avoids unnecessary rounding. A 30°–60°–90° triangle has side ratio 1:√3:2, while a 45°–45°–90° triangle has ratio 1:1:√2. Draw which side matches which angle rather than memorising three numbers without their positions.
A full turn is 360°, while a radian is the angle subtended by an arc equal in length to the radius. Thus radians. School triangle problems commonly use degrees, but calculus formulas for trigonometric derivatives conventionally use radians. A calculator set to the wrong mode can produce a plausible-looking wrong answer. Test the mode with sin 30° = 0.5 when using degrees. Convert an angle only when needed and label the unit consistently.
Extend ratios around the unit circle
On a unit circle, a point reached by angle θ has coordinates (cos θ, sin θ). This explains why sine and cosine can be negative outside the first quadrant. In quadrant two sine is positive and cosine negative; in quadrant three both are negative; in quadrant four sine is negative and cosine positive. Tangent equals sin θ/cos θ where cos θ ≠ 0. Therefore tan θ is undefined at 90° and 270°. Reference angles help evaluate expressions such as.
The identity sin²θ + cos²θ = 1 follows from Pythagoras on the unit circle. It does not imply sin θ + cos θ = 1. Sine and cosine have period 360° and range −1 to 1. Sine begins at zero when θ = 0°, while cosine begins at 1. For, amplitude is 3. For with degree input, the period is 180°. Sketch key points before connecting them smoothly, and keep the angle scale even.
Elevation, depression and bearings
An angle of elevation is measured upward from the observer's horizontal line. An angle of depression is measured downward from a horizontal line at the higher observer. With parallel horizontal lines, the depression angle equals the corresponding elevation angle, but neither is necessarily the angle between the sight line and a vertical wall. If an observer's eyes are 1.5 m above the ground, add that eye height after finding the vertical rise above the eye level.
A three-figure bearing is measured clockwise from north. East is 090°, south 180° and west 270°. The reverse bearing differs by 180°, adjusted into the interval from 000° up to but not including 360°. Thus the reverse of 065° is 245°, and the reverse of 250° is 070°. At each point, draw a north line. When two journeys are combined, the angle inside the travel triangle may be supplementary to a difference of bearings, so derive it from the north lines instead of subtracting numbers mechanically.
Solve triangles without a right angle
For a triangle with sides a, b and c opposite angles A, B and C, the sine rule is. It is useful when a known side has a known opposite angle. The cosine rule is and is useful for two sides with their included angle or for all three sides. For a = 5, b = 7 and C = 60°, c² = 25 + 49 − 35 = 39, so.
The area formula uses the included angle between sides a and b. It reduces to the ordinary half-base-times-height formula when the perpendicular height is b sin C. If you know two sides and a non-included angle and use the sine rule, two possible triangles can sometimes fit. Since, inspect the supplementary angle and whether all three angles still sum to 180°. An inverse-sine calculator output alone may not give every valid solution.
Check a complete solution
Self-check: find sin 150°, the reverse bearing of 035°, and the area of a triangle with sides 8 cm and 5 cm enclosing 30°. The answers are 1/2, 215°, and 10 cm². A sensible length must be positive, triangle angles must total 180°, and the longest side must face the largest angle. Leave exact surds or stated π expressions when requested. For an approximate result, carry guard digits through the calculation and round only the final value to the required accuracy. These checks catch a wrong ratio, an incorrectly labelled included angle, or a calculator-mode error before it becomes the final answer.
Build further exact angles from angle sums
The addition formulas include and cos(A + B) = cos A cos B − sin A sin B. Thus. For a difference. These are not the same as adding the sine values of the two angles.
When solving a trigonometric equation over a stated interval, list every permitted solution. For on 0° ≤ θ ≤ 360°, both 30° and 150° work. For cos θ = 0, solutions are 90° and 270°. Check boundaries individually if they are included, and use the function's period to find further solutions only within the requested interval. A single inverse-calculator value is often just a starting point.
Tangent has period 180°, so tan 30° and tan 210° agree where both are defined. Its periodic pattern differs from the 360° period of sine and cosine.
Include negative angles and all required quadrants
A negative angle turns clockwise from the positive horizontal direction on the unit circle. Reflection across the horizontal axis gives and; consequently where defined. Thus. To solve within, include negative representatives and inspect both endpoints. For sin θ = 0 on that interval, the solutions are −360°, −180°, 0°, 180° and 360°.
Exact values for other special angles follow from reference angles or addition formulas. For, sine is √2/2 and cosine −√2/2. Since 105° and 75° are supplementary. Their cosines have opposite signs. A value and its sign both matter: finding the correct reference-angle magnitude while choosing the wrong quadrant still gives an incorrect answer.