In the triangle shown, what is the value of tan y°?

1. TRANSLATE the diagram information

From the diagram, we can identify:

• Horizontal top side: length = 15
• Vertical right side: length = 20
• Right angle location: top-right corner
• Angle of interest: \(\mathrm{y°}\) at the bottom vertex

2. INFER which sides are opposite and adjacent to angle \(\mathrm{y°}\)

This is the crucial step! We need to identify the sides relative to angle \(\mathrm{y°}\).

Think about angle \(\mathrm{y°}\) sitting at the bottom vertex. Ask yourself:

• Which side is across from angle \(\mathrm{y°}\) (not touching it)? That's the horizontal top side with length 15. This is the opposite side.
• Which side touches angle \(\mathrm{y°}\) (besides the hypotenuse)? That's the vertical side with length 20. This is the adjacent side.

Key insight: The opposite side is always across from the angle, and the adjacent side always touches the angle.

3. Apply the tangent formula

Now that we know:

• Opposite = 15
• Adjacent = 20

We can use the tangent definition:

\(\tan \mathrm{y°} = \frac{\mathrm{opposite}}{\mathrm{adjacent}} = \frac{15}{20}\)

4. SIMPLIFY the fraction

\(\frac{15}{20} = \frac{3}{4} = 0.75\)

Answer: 0.75 (also acceptable: 3/4 or 0.750)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing which side is opposite vs adjacent to angle \(\mathrm{y°}\)

Many students struggle with spatial reasoning in triangles, especially when the triangle isn't in the "standard" orientation (with the right angle at bottom-left). They might look at the triangle and think:

• "The vertical side is 20, and vertical seems 'opposite' to me"
• "The horizontal side is 15, so that's adjacent"

This flawed spatial reasoning leads them to calculate:

\(\tan \mathrm{y°} = \frac{20}{15} = \frac{4}{3} \approx 1.33\)

While 1.33 isn't one of the typical multiple choice options in this problem type, this error causes them to arrive at a completely different answer and leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion: Mixing up the tangent ratio with sine or cosine

Some students remember they need a trigonometric ratio but can't recall which one. They might think:

• \(\tan \mathrm{y°} = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\) (this is actually sine)
• \(\tan \mathrm{y°} = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\) (this is actually cosine)

To use these incorrectly, they'd first need to calculate the hypotenuse using the Pythagorean theorem:

\(\mathrm{hypotenuse} = \sqrt{15^2 + 20^2}\)
\(= \sqrt{225 + 400}\)
\(= \sqrt{625}\)
\(= 25\)

Then they might calculate:

• \(\sin \mathrm{y°} = \frac{15}{25} = 0.6\), or
• \(\cos \mathrm{y°} = \frac{20}{25} = 0.8\)

This conceptual confusion causes them to arrive at incorrect values and leads to guessing.

The Bottom Line:

This problem tests whether students can correctly identify opposite and adjacent sides relative to a specific angle in a right triangle, regardless of the triangle's orientation. The key is to always reference sides relative to the angle in question, not based on how the triangle "looks" on the page.