Grade 11 Gen Math | Trigonometric Ratio in Right Angled Triangle Second Term (Term 2) Week 3 Part 1

Summary

The lesson introduces trigonometric ratios for right triangles and explains how to use them to find missing sides and angles, including in practical situations.

Core concepts

A trigonometric ratio compares the lengths of sides in a right triangle in relation to one of its acute angles.

The sides are identified relative to the chosen angle:

• Opposite: Across from the chosen angle.
• Adjacent: Next to the chosen angle, but not the hypotenuse.
• Hypotenuse: Across from the right angle. It remains the same regardless of which acute angle is chosen.

The labels opposite and adjacent depend on the reference angle; the hypotenuse does not.

The six trigonometric ratios

The lesson uses SOHCAHTOA to remember the first three ratios:

• Sine: (\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}})
• Cosine: (\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}})
• Tangent: (\tan\theta = \frac{\text{opposite}}{\text{adjacent}})

The other three ratios are their reciprocals:

• Cosecant: (\csc\theta = \frac{\text{hypotenuse}}{\text{opposite}})
• Secant: (\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}})
• Cotangent: (\cot\theta = \frac{\text{adjacent}}{\text{opposite}})

These ratios are used with right triangles.

Using a calculator

• Set the calculator to degree mode when working with angles measured in degrees.
• To evaluate a ratio such as (\cos 80^\circ), enter the trig function and angle, then round as requested.
• To find an angle from a ratio, use the inverse function, often accessed with SHIFT, 2nd function, or an inverse option:
  • (\theta = \cos^{-1}(x)), (\sin^{-1}(x)), or (\tan^{-1}(x)), as appropriate.
• Round angle answers to the requested precision. For answers to the nearest minute, convert the decimal-degree result to degrees and minutes using the calculator’s DMS function.

General method for solving missing parts

1. Draw or inspect the right triangle and mark the right angle.
2. Identify the reference angle relevant to the unknown.
3. Label the opposite, adjacent, and hypotenuse sides relative to that angle.
4. Select the ratio that connects the known information to the unknown:
   • Use sine for the opposite and hypotenuse.
   • Use cosine for the adjacent and hypotenuse.
   • Use tangent for the opposite and adjacent.
5. Substitute the known values and solve:
   • For an unknown side, rearrange the equation or cross-multiply.
   • For an unknown angle, use the appropriate inverse trig function.
6. If two sides are known and the third side is needed, use the Pythagorean theorem rather than a trigonometric ratio.
7. In a right triangle, the two acute angles add to (90^\circ), so one can be found by subtracting the other from (90^\circ).
8. Round the result as requested and include the correct unit.

Worked examples and applications

The instructor demonstrates how to write equations from triangle diagrams and use them to find missing sides and angles. The lesson also applies the same process to three real-world problems:

• Kite: A 30 m string forms a (42^\circ) angle with the ground. The height is found using sine, giving about 20.07 m.
• Ladder: A ladder rests against an 8 m wall and forms a (65^\circ) angle with the ground. Its length is found using sine, giving about 8.827 m.
• Wheelchair ramp: A 5 m ramp rises 1.25 m. The angle of inclination is found using inverse sine, giving approximately 14.48°. Although the captions give the final unit as meters, the answer is an angle and should be expressed in degrees.

The lesson concludes with practice activities and a multiple-choice check. Reported answers include (P = 13.95) units, (Q = 5) units, and (R \approx 30.96^\circ) (about (31^\circ)). The quiz reinforces SOHCAHTOA, choosing a ratio based on side positions, and interpreting a right-triangle diagram.

Speakers and sources featured

• Instructor/narrator: The presenter of Native Man Math Tutorial, delivering the lesson.
• Other speakers: None featured.