For an acute angle θ, it is given that θ = (9)/(41) and θ = (40)/(41). What is the exact value of θ?

(9)/(40). Using the quotient identity θ = ( θ)/( θ), we divide (9)/(41) by (40)/(41). Dividing by a fraction is the same as multiplying by its reciprocal, which gives (9)/(41) × (41)/(40) = (9)/(40).

Grade 10 · Core Mathematics · Lesson 6 of 11

Relating Sine, Cosine, and Tangent

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners derive the quotient identity θ = ( θ)/( θ) for acute angles through empirical investigation and side-ratio division.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L640 minutes · 2 slides

KICD learning outcome (c)

Relate the sine, cosine and tangent of acute angles.

By the end of this lesson, learners should be able to

Derive the quotient identity $\tan θ = \frac{\sin θ}{\cos θ}$ for acute angles through empirical investigation and side-ratio division.
Apply and rearrange the trigonometric identity $\tan θ = \frac{\sin θ}{\cos θ}$ to determine missing trigonometric ratios in exact fraction and surd forms.

Key vocabulary

Quotient identity: A trigonometric equation expressing one ratio as the division of two other ratios.; The quotient identity $\tan θ = \frac{\sin θ}{\cos θ}$ relates tangent to sine and cosine.
Acute angle: An angle whose measure is strictly greater than zero and less than ninety degrees.; In right-angled triangle calculations, we evaluate trigonometric ratios for an acute angle $θ$ .
Reciprocal: The multiplicative inverse of a number, obtained by swapping its numerator and denominator.; Multiplying $\frac{\sqrt{2}}{2}$ by its reciprocal $\frac{2}{\sqrt{2}}$ yields $1$ .

Lesson preview

Slide 1 · Discovering the Relationship Between Sine, Cosine, and TangentOpening · 18 min

Figure 1: A right-angled triangle and a completed table showing the sine, cosine, and tangent ratios for four acute angles.

For any acute angle $θ$ in a right-angled triangle, the sine ratio is the opposite side divided by the hypotenuse, while the cosine ratio is the adjacent side divided by the hypotenuse. When we divide the sine ratio by the cosine ratio, the hypotenuse cancels out, leaving the opposite side divided by the adjacent side. This ratio is precisely the tangent of the angle. Therefore, for every acute angle $θ$ , the fundamental quotient identity $\tan θ = \frac{\sin θ}{\cos θ}$ holds true.

Dividing the opposite over hypotenuse ratio by adjacent over hypotenuse yields opposite over adjacent.
The trigonometric quotient identity states that tangent equals sine divided by cosine for acute angles.
Calculating the quotient of sine and cosine values gives the exact tangent ratio of that angle.
This relationship connects all three primary trigonometric ratios for any acute angle $θ$ .

Worked example. State the quotient identity:

\tan β = \frac{\sin β}{\cos β}

. Substitute the given trigonometric values into the equation:

\tan β = \frac{0.6}{0.8}

. Simplify the fraction:

\frac{0.6}{0.8} = \frac{6}{8} = \frac{3}{4} = 0.75

. Answer: $\tan β = 0.75$ .

Check question: For an acute angle $θ$ , it is given that $\sin θ = \frac{9}{41}$ and $\cos θ = \frac{40}{41}$ . What is the exact value of $\tan θ$ ?

$\frac{9}{40}$
$\frac{40}{9}$
$\frac{360}{1681}$
$\frac{49}{41}$

Show answer

$\frac{9}{40}$ . Using the quotient identity $\tan θ = \frac{\sin θ}{\cos θ}$ , we divide $\frac{9}{41}$ by $\frac{40}{41}$ . Dividing by a fraction is the same as multiplying by its reciprocal, which gives $\frac{9}{41} \times \frac{41}{40} = \frac{9}{40}$ .

Slide 2 · Applying and Rearranging the Quotient IdentityPractice · 22 min

Figure 2: A diagram showing the algebraic rearrangement of the quotient identity alongside a step-by-step worked solution using surds.

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Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Relating Sine, Cosine, and Tangent

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Trigonometry 1