Road to 800

Trigonometry

Chapter 14 · 2 lessons · 56 questions

SOH-CAH-TOA and cofunctions

Key points

  • sin⁡\sin =oppositehypotenuse= \dfrac{\text{opposite}}{\text{hypotenuse}}, cos⁡\cos =adjacenthypotenuse= \dfrac{\text{adjacent}}{\text{hypotenuse}} and tan⁡\tan =oppositeadjacent= \dfrac{\text{opposite}}{\text{adjacent}}.
  • ‘Opposite’ and ‘adjacent’ depend on which angle you are looking from.
  • The two acute angles of a right triangle add to 90∘90^\circ, so sin⁡x∘\sin x^\circ =cos⁡(90∘−x∘)= \cos(90^\circ - x^\circ).
  • Similar right triangles have the same ratios.

See it solved

Example

In the figure, triangle ABCABC has a right angle at CC, with AB=17AB = 17 and BC=8BC = 8. What is the value of cos⁡A\cos A?

ACB178

Worked by hand

The plan

  1. Determine the ratio of sides
  2. Determine the ratio asked for

The ratio comes from the triangle, by hand. Desmos is a check: it finds the missing side and shows the ratio as a decimal.

  1. Step 1: Determine the ratio of sides

    • Look from angle AA. The hypotenuse is AB=17AB = 17, and the opposite side is BC=8BC = 8.
    • The adjacent side is ACAC, which is missing.
    • 88-1515-1717 is a triple, so AC=15AC = 15.
    • Check: 82+1528^2 + 15^2 =64+225= 64 + 225 =289= 289, and 172=28917^2 = 289.

  2. Step 2: Determine the ratio asked for

    • cos⁡A\cos A =adjacenthypotenuse= \dfrac{\text{adjacent}}{\text{hypotenuse}} =1517= \dfrac{15}{17}.
The correct answer is 1517\dfrac{15}{17}.

The same example in Desmos

  1. new graph

    Keys: sqrt 17^2 → -8^2

    1515: the missing side ACAC.

  2. About 0.8820.882: cos⁡A\cos A as a decimal, in case the choices are decimals.

Type
Desmos expression list: sqrt{17^2-8^2}; frac{15}{17}

Made with Desmos

Open Desmos

Your turn

In a right triangle, sin⁡32∘=cos⁡x∘\sin 32^\circ = \cos x^\circ, where 0<x<900 < x < 90. What is the value of xx?

Show the answerHide the answer

Answer

The correct answer is 5858.

  1. Step 1: Use the cofunction rule

    • The sine of an angle equals the cosine of the other acute angle in the triangle.
    • The two acute angles add to 90∘90^\circ, so the other angle is 90−32=5890 - 32 = 58 degrees.
    • So x=58x = 58.

Traps

  • Mixing up opposite and adjacent. From angle AA, the side BC=8BC = 8 is opposite. 817\dfrac{8}{17} is sin⁡A\sin A, not cos⁡A\cos A.
  • Using a leg as the hypotenuse. The hypotenuse is always opposite the right angle: here it is AB=17AB = 17. 158\dfrac{15}{8} cannot be a cosine, because a cosine is never more than 11.
Practice this lesson40 questions

The unit circle

Key points

  • The unit circle has its center at (0,0)(0, 0) and a radius of 11.
  • The point at angle θ\theta is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta): cosine is the xx-coordinate and sine is the yy-coordinate.
  • The signs come from the quadrant the angle is in.
  • A full turn is 360∘360^\circ, or 2π2\pi radians. Adding a full turn does not change the point.

See it solved

Example

What is the value of cos⁡(2π3)\cos\left(\dfrac{2\pi}{3}\right)?

Worked by hand

The plan

  1. Locate the angle
  2. Use the reference angle

Desmos is quicker as a calculator here: check the angle mode, then type the value. Rule out choices by their sign first.

  1. Step 1: Locate the angle

    • π\pi radians is 180∘180^\circ, so 2π3\dfrac{2\pi}{3} radians is 2⋅1803=120\dfrac{2 \cdot 180}{3} = 120 degrees.
    • 120∘120^\circ is in the second quadrant, where xx is negative. So the cosine is negative.

  2. Step 2: Use the reference angle

    • 180−120=60180 - 120 = 60. The reference angle is 60∘60^\circ.
    • cos⁡60∘=12\cos 60^\circ = \dfrac{1}{2}.
    • With the sign, cos⁡(2π3)=−12\cos\left(\dfrac{2\pi}{3}\right) = -\dfrac{1}{2}.
The correct answer is −12-\dfrac{1}{2}.

The same example in Desmos

  1. wrench, then Radians

    −0.5-0.5, at the right of the row. That is −12-\dfrac{1}{2}.

Type
Desmos expression list: cos(frac{2pi}{3})

Made with Desmos

Open Desmos

Your turn

The point on the unit circle at the angle θ\theta is (−35,45)\left(-\dfrac{3}{5}, \dfrac{4}{5}\right). What is the value of tan⁡θ\tan\theta?

Show the answerHide the answer

Answer

The correct answer is −43-\dfrac{4}{3}.

  1. Step 1: Read the sine and the cosine from the point

    • cos⁡θ\cos\theta is the xx-coordinate: −35-\dfrac{3}{5}.
    • sin⁡θ\sin\theta is the yy-coordinate: 45\dfrac{4}{5}.

  2. Step 2: Determine the tangent

    • tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}, which is 45\dfrac{4}{5} divided by −35-\dfrac{3}{5}.
    • Dividing by a fraction is multiplying by its flip: 45⋅(−53)=−2015\dfrac{4}{5} \cdot \left(-\dfrac{5}{3}\right) = -\dfrac{20}{15}.
    • −2015=−43-\dfrac{20}{15} = -\dfrac{4}{3}.

Traps

  • Dropping the sign. The reference angle gives 12\dfrac{1}{2}, but 120∘120^\circ is in the second quadrant, where the cosine is negative. 12\dfrac{1}{2} is a wrong answer.
  • Swapping sine and cosine. Cosine is the xx-coordinate. 32\dfrac{\sqrt{3}}{2} is the yy-coordinate at this angle, which is the sine.
Practice this lesson16 questions
Typing tips for Desmos
  • In the keys lines, → means the right-arrow key.
  • Check the angle mode before any trig: the wrench (Graph Settings) has Radians and Degrees. sin(30) = 0.5 means you are in degrees.
  • Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
  • Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
Start practicing

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