Trigonometry

The One Thing

In a right-angled triangle, trigonometry relates a selected acute angle to the opposite side, adjacent side, and hypotenuse through sine, cosine, and tangent. Correct solutions depend on labelling the triangle relative to the chosen angle, selecting the appropriate ratio, and verifying the result where possible.

Definitions and Results

• Right-angled triangle: A triangle containing one angle equal to 90 degrees.
• Hypotenuse: The side opposite the right angle; it is the longest side of a right-angled triangle.
• Opposite side: For a selected acute angle, the side directly opposite that angle.
• Adjacent side: For a selected acute angle, the side next to the angle other than the hypotenuse.
• Trigonometric ratio: A comparison of two sides of a right-angled triangle with respect to a chosen acute angle.
• Sine: For an acute angle theta,

sin theta = opposite side divided by hypotenuse.

• Cosine: For an acute angle theta,

cos theta = adjacent side divided by hypotenuse.

• Tangent: For an acute angle theta,

tan theta = opposite side divided by adjacent side.

• Cosecant: The reciprocal of sine,

cosec theta = 1 divided by sin theta = hypotenuse divided by opposite side.

• Secant: The reciprocal of cosine,

sec theta = 1 divided by cos theta = hypotenuse divided by adjacent side.

• Cotangent: The reciprocal of tangent,

cot theta = 1 divided by tan theta = adjacent side divided by opposite side.

• Reciprocal relationships:

sin theta multiplied by cosec theta = 1, cos theta multiplied by sec theta = 1, and tan theta multiplied by cot theta = 1.

• Fundamental identities:

sin squared theta plus cos squared theta = 1, 1 plus tan squared theta = sec squared theta, and 1 plus cot squared theta = cosec squared theta.

• Pythagorean theorem:

hypotenuse squared = perpendicular squared + base squared.

• Angle of elevation: The angle formed between the horizontal line of sight and the line of sight to an object above the observer.
• Angle of depression: The angle formed between the horizontal line through the observer and the line of sight to an object below the observer.
• Complementary angles: Two angles whose sum is 90 degrees. The two acute angles of a right-angled triangle are complementary.
• Complementary-angle relationships:

sin theta = cos of (90 degrees minus theta), cos theta = sin of (90 degrees minus theta), tan theta = cot of (90 degrees minus theta), and vice versa.

• Standard sine values:

sin 0 degrees = 0, sin 30 degrees = 1/2, sin 45 degrees = 1 divided by square root of 2, sin 60 degrees = square root of 3 divided by 2, sin 90 degrees = 1.

• Standard cosine values:

cos 0 degrees = 1, cos 30 degrees = square root of 3 divided by 2, cos 45 degrees = 1 divided by square root of 2, cos 60 degrees = 1/2, cos 90 degrees = 0.

• Standard tangent values:

tan 0 degrees = 0, tan 30 degrees = 1 divided by square root of 3, tan 45 degrees = 1, tan 60 degrees = square root of 3, while tan 90 degrees is not defined.

• Reference-angle dependence: The hypotenuse is always opposite the right angle, but the opposite and adjacent sides change when the reference angle changes.
• Use of trigonometry: Trigonometric ratios connect angles, sides, heights, and distances in right-angled triangles. Distances and heights must use consistent units, and final answers should include suitable units where appropriate.

Worked Methods

1. Selecting and applying a trigonometric ratio

• Draw the right-angled triangle.
• Mark the selected acute reference angle.
• Identify the opposite side, adjacent side, and hypotenuse relative to that angle.
• Identify the known and unknown quantities.
• Select the suitable ratio:

- Use sin theta = opposite side divided by hypotenuse when the opposite side and hypotenuse are involved. - Use cos theta = adjacent side divided by hypotenuse when the adjacent side and hypotenuse are involved. - Use tan theta = opposite side divided by adjacent side when the opposite and adjacent sides are involved.

• Substitute the given values.
• Rearrange if necessary and simplify.
• Include suitable units in the final answer.

The reciprocal ratios may also be used:

• cosec theta = hypotenuse divided by opposite side;
• sec theta = hypotenuse divided by adjacent side;
• cot theta = adjacent side divided by opposite side.

2. Finding an angle from a known ratio

• Draw and label the right-angled triangle.
• Identify the selected acute angle and the relevant sides.
• Form the appropriate ratio using sine, cosine, or tangent.
• Apply the corresponding inverse trigonometric function:

- sin inverse for a sine ratio; - cos inverse for a cosine ratio; - tan inverse for a tangent ratio.

• State the angle in degrees where appropriate.
• Check that the result is an acute angle of the right-angled triangle.

3. Solving practical problems involving elevation and depression

• Draw a clear diagram of the situation.
• Represent the horizontal line and the line of sight.
• Identify whether the problem gives an angle of elevation or an angle of depression.
• Mark the right angle and label the height, horizontal distance, and line of sight.
• Select the reference angle and identify the opposite side, adjacent side, and hypotenuse.
• Choose the appropriate trigonometric ratio.
• Substitute the known measurements and solve for the unknown height, distance, or angle.
• Keep all distances and heights in consistent units.
• Include suitable units in the final answer.

4. Verifying a right-triangle solution

• Use the Pythagorean theorem:

hypotenuse squared = perpendicular squared + base squared.

• Substitute the three side lengths.
• Check that the equality holds.
• Where suitable, verify the result using a fundamental identity, such as:

sin squared theta plus cos squared theta = 1, 1 plus tan squared theta = sec squared theta, or 1 plus cot squared theta = cosec squared theta.

5. Using complementary angles

• Identify the two acute angles of the right-angled triangle.
• Confirm that their sum is 90 degrees.
• Use the complementary-angle relationships:

- sin theta = cos of (90 degrees minus theta); - cos theta = sin of (90 degrees minus theta); - tan theta = cot of (90 degrees minus theta); - and the corresponding reverse relationships.

• Substitute the known ratio or angle and simplify.

Where It Goes Wrong

• The reference angle is not identified before labelling the opposite and adjacent sides. The hypotenuse remains opposite the right angle, but the opposite and adjacent sides change when the reference angle changes.
• The wrong trigonometric ratio is selected because the known and unknown sides are not compared carefully.
• The hypotenuse is confused with the adjacent side. The hypotenuse is always the side opposite the right angle.
• A reciprocal ratio is used without applying the correct relationship: cosec theta = 1 divided by sin theta, sec theta = 1 divided by cos theta, and cot theta = 1 divided by tan theta.
• The conditions of the identities are forgotten. These ratios and identities are being applied to a right-angled triangle and a selected acute angle; tan 90 degrees is not defined.
• Practical problems are solved without a clear diagram or with inconsistent units for distances and heights.

What Gets Asked

• Define the hypotenuse, opposite side, adjacent side, right-angled triangle, and trigonometric ratio.
• State or use the definitions of sin theta, cos theta, and tan theta.
• State or use the reciprocal ratios cosec theta, sec theta, and cot theta.
• Prove or apply the reciprocal relationships and the fundamental identities.
• Use the Pythagorean theorem to find a missing side or verify a solution.
• Calculate an unknown side by drawing and labelling a right-angled triangle, selecting a ratio, substituting values, and simplifying.
• Calculate an unknown angle using sin inverse, cos inverse, or tan inverse.
• Use the standard values of sine, cosine, and tangent for 0 degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees, including the fact that tan 90 degrees is not defined.
• Apply complementary-angle relationships such as sin theta = cos of (90 degrees minus theta) and tan theta = cot of (90 degrees minus theta).
• Solve practical problems involving heights, distances, angles of elevation, and angles of depression.
• Check whether a solution satisfies the Pythagorean theorem or an appropriate trigonometric identity.