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Let's break down the problem step by step:

### Given:
- \( AB = 15 \, \text{cm} \)
- \( AC = 17 \, \text{cm} \)
- \( CD = 12 \, \text{cm} \)
- \( AB \perp BD \) (meaning \( \triangle ABD \) is a right-angled triangle at \( B \))

We need to find the following trigonometric ratios **without evaluating any angle**:

#### (a) \( \tan \angle ACD \)

We can use the definition of tangent:
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\]
For \( \angle ACD \), the opposite side is \( CD \), and the adjacent side is \( AC \):
\[
\tan \angle ACD = \frac{CD}{AC} = \frac{12}{17}
\]

#### (b) \( \sin \angle ACD \)

The sine of an angle is the ratio of the opposite side to the hypotenuse:
\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
\]
For \( \angle ACD \), the opposite side is \( CD \), and the hypotenuse is \( AD \). To find \( AD \), we can apply the Pythagorean theorem in \( \triangle ABD \):

\[
AD = \sqrt{AB^2 + BD^2} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25
\]

So,
\[
\sin \angle ACD = \frac{CD}{AD} = \frac{12}{25}
\]

#### (c) \( \cos \angle ADC \)

The cosine of an angle is the ratio of the adjacent side to the hypotenuse. For \( \angle ADC \), the adjacent side is \( AC \), and the hypotenuse is \( AD \):
\[
\cos \angle ADC = \frac{AC}{AD} = \frac{17}{25}
\]

#### (d) \( \sin \angle CAD \)

For \( \angle CAD \), the sine is the ratio of the opposite side \( CD \) to the hypotenuse \( AD \):
\[
\sin \angle CAD = \frac{AB}{AC} = \frac{8}{17}
\]

These are your trigonometric ratios based on the given triangle and dimensions.

Do you need any further details or explanations? Here are 5 related questions to consider:

1. How do you find the value of \( \cos \angle ACD \) from the information given?
2. What is the significance of the Pythagorean theorem in solving right-angled triangles like this one?
3. Can you calculate the area of \( \triangle ACD \)?
4. How would the ratios change if \( CD \) were increased by 5 cm?
5. What is the relationship between the sine and cosine of complementary angles in a triangle?

**Tip:** Always double-check which side is opposite and adjacent when dealing with trigonometric functions, particularly in non-right-angled triangles.

In the diagram, BCD is a straight line and AB is perpendicular to BD. AB = 15 cm, AC = 17 cm, CD = 12 cm. Without evaluating any angle, find: (a) tan∠ACD, (b) sin∠ACD, (c) cos∠ADC, (d) sin∠CAD.

Solve for trigonometric ratios such as tan, sin, and cos for a right-angled triangle where AB = 15 cm, AC = 17 cm, and CD = 12 cm. This problem involves the Pythagorean theorem and basic trigonometric formulas. Suitable for students in Grades 10-12.

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