Guided Practice
Step-by-step practice problems to reinforce your understanding of trigonometric ratios and solving right triangles.
How to Use This Lesson
This guided practice lesson provides structured problems with hints and step-by-step solutions. Work through each problem carefully before checking the solution. Remember:
- SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
- Always identify which angle you're working with first
- Label the sides relative to your reference angle
- Choose the appropriate ratio based on what you know and what you need to find
Worked Examples
Example 1: Finding a Missing Side Using Sine
Problem: In right triangle ABC, angle A = 35 degrees, the hypotenuse (c) = 12 cm. Find the length of side a (opposite to angle A).
Show Solution
Step 1: Identify what we know and what we need.
- We know: angle A = 35 degrees, hypotenuse = 12 cm
- We need: opposite side (a)
Step 2: Choose the right ratio. We have the hypotenuse and need the opposite, so use sine.
sin(A) = opposite / hypotenuse
Step 3: Substitute and solve.
sin(35) = a / 12
a = 12 * sin(35)
a = 12 * 0.5736
Answer: a = 6.88 cm
Example 2: Finding a Missing Angle Using Inverse Tangent
Problem: In a right triangle, the side opposite to angle X is 8 units and the adjacent side is 15 units. Find angle X.
Show Solution
Step 1: Identify what we know.
- Opposite = 8 units
- Adjacent = 15 units
Step 2: Since we have opposite and adjacent, use tangent.
tan(X) = opposite / adjacent = 8 / 15
Step 3: Use inverse tangent to find the angle.
X = tan^(-1)(8/15)
X = tan^(-1)(0.5333)
Answer: X = 28.07 degrees
Practice Problems
Work through these problems. Each includes a hint if you get stuck.
Problem 1
In right triangle PQR, angle P = 42 degrees and the hypotenuse = 20 cm. Find the length of the side adjacent to angle P.
Hint
Use cosine since you have the hypotenuse and need the adjacent side. cos(P) = adjacent / hypotenuse
Answer
adjacent = 20 * cos(42) = 20 * 0.7431 = 14.86 cm
Problem 2
A ladder leans against a wall making a 65-degree angle with the ground. If the ladder is 18 feet long, how high up the wall does it reach?
Hint
The ladder is the hypotenuse, and the height on the wall is opposite to the 65-degree angle. Use sine.
Answer
height = 18 * sin(65) = 18 * 0.9063 = 16.31 feet
Problem 3
Find angle B in a right triangle where the opposite side is 7 cm and the hypotenuse is 25 cm.
Hint
Use inverse sine: sin(B) = opposite / hypotenuse, then B = sin^(-1)(opposite/hypotenuse)
Answer
sin(B) = 7/25 = 0.28, B = sin^(-1)(0.28) = 16.26 degrees
Problem 4
In a right triangle, if tan(A) = 3/4, what is the value of sin(A)?
Hint
If tan(A) = 3/4, then opposite = 3 and adjacent = 4. Use the Pythagorean theorem to find the hypotenuse, then calculate sin(A).
Answer
hypotenuse = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. sin(A) = 3/5 = 0.6
Problem 5
A ramp rises 3 meters over a horizontal distance of 12 meters. What angle does the ramp make with the ground?
Hint
The rise is opposite and the horizontal distance is adjacent. Use inverse tangent.
Answer
angle = tan^(-1)(3/12) = tan^(-1)(0.25) = 14.04 degrees
Problem 6
In right triangle XYZ with right angle at Y, if angle X = 58 degrees and side XZ (hypotenuse) = 15 units, find side YZ.
Hint
Side YZ is opposite to angle X. Use sine: sin(58) = YZ / 15
Answer
YZ = 15 * sin(58) = 15 * 0.8480 = 12.72 units
Problem 7
A tree casts a shadow 45 feet long. The angle of elevation from the tip of the shadow to the top of the tree is 33 degrees. How tall is the tree?
Hint
The shadow is adjacent to the angle, and the tree height is opposite. Use tangent.
Answer
height = 45 * tan(33) = 45 * 0.6494 = 29.22 feet
Problem 8
If cos(theta) = 5/13, find tan(theta). Assume theta is an acute angle.
Hint
If cos(theta) = adjacent/hypotenuse = 5/13, then adjacent = 5 and hypotenuse = 13. Find opposite using Pythagorean theorem.
Answer
opposite = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12. tan(theta) = 12/5 = 2.4
Problem 9
A helicopter is flying at an altitude of 500 meters. The angle of depression to a landing pad is 12 degrees. How far horizontally is the helicopter from the landing pad?
Hint
The angle of depression equals the angle of elevation from the pad. The altitude (500m) is opposite, and you need the adjacent (horizontal distance). Use tangent.
Answer
tan(12) = 500 / horizontal. horizontal = 500 / tan(12) = 500 / 0.2126 = 2352.2 meters
Problem 10
In a right triangle, the two legs measure 9 cm and 12 cm. Find all three angles of the triangle.
Hint
One angle is 90 degrees (the right angle). Use inverse tangent to find one acute angle, then subtract from 90 to find the other.
Answer
Let angle A be opposite the 9 cm side. tan(A) = 9/12 = 0.75, A = tan^(-1)(0.75) = 36.87 degrees. The other acute angle = 90 - 36.87 = 53.13 degrees. Angles: 36.87, 53.13, and 90 degrees
Check Your Understanding
Answer these questions to verify your understanding:
- When should you use sine vs. cosine vs. tangent?
- How do you find an angle when you know two sides?
- What is the relationship between angle of elevation and angle of depression?
- If you know sin(A), how can you find cos(A) without knowing the angle?
Check Your Answers
- Use sine when you have opposite and hypotenuse, cosine when you have adjacent and hypotenuse, and tangent when you have opposite and adjacent.
- Use the inverse function (sin^(-1), cos^(-1), or tan^(-1)) of the ratio of the two known sides.
- They are equal; the angle of depression from point A to point B equals the angle of elevation from point B to point A.
- Use the Pythagorean identity: sin^2(A) + cos^2(A) = 1, so cos(A) = sqrt(1 - sin^2(A)).
Next Steps
- If you scored 8/10 or higher on the practice problems, move on to Word Problems
- If you struggled with finding angles, review the inverse trig functions
- If you had trouble setting up problems, review how to identify opposite, adjacent, and hypotenuse
- Try creating your own problems and solving them for extra practice