Finding Angles in Physics

Trigonometry · Axiom Academy

LESSON Finding Launch Angles in Projectile Motion Discover why two different launch angles can achieve the same range—and how trigonometry reveals this hidden symmetry. Imagine you're launching a projectile (like a cannonball, basketball, or water from a fountain) and you want it to land at a specific distance. You know the initial velocity, but what launch angle should you use? The Surprise: For most ranges, there are actually two different angles that will get you to the same target! Adjust the launch angle and watch how it affects the range. Try to find the angle that gives the maximum distance! Let's say you want to hit a target at 86.6 meters with an initial velocity of 30 m/s . Watch what happens when we test different angles! Lower trajectory, faster horizontal speed Higher trajectory, longer flight time What do you notice about these two angles? The range formula for projectile motion uses a double angle identity from trigonometry: The key is the sin(2θ) term. Remember from trigonometry: But sin(120°) = sin(180° - 120°) = sin(60°) The Pattern: If angles θ₁ and θ₂ are complementary (add to 90°), then: sin(2θ₁) = sin(2θ₂) Therefore, they produce the same range ! Understanding the two-angle phenomenon is crucial in many fields: Athletes and coaches choose between a low, fast trajectory or a high, arcing shot depending on obstacles and defenders.

This is the written version of the interactive lesson above. See the full Trigonometry course.