1. Aim: How do solve related rate problems involving
trig functions?
Do Now: 1. Evaluate the following:
a. sin 𝜋 4
b. tan 𝜋 4
c. sec 𝜋 3
2. Differentiate the following:
𝑎. tan 𝜃
𝑏. sec 𝜃

2. Mario is standing 25 feet from the launch pad of a hot air balloon
Mario is standing 25 feet from the launch pad of a hot air balloon. If the balloon is rising at the rate of 20 ft/min, how will the angle between Mario’s eye and the balloon be changing after 5 minutes?
𝑑𝑦 𝑑𝑡 =20
𝑡𝑎𝑛𝜃= 𝑦 25 z
𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 𝑑𝑦 𝑑𝑡
When t = 5,
y = 5(20) = 100 y 25 𝜃
𝑠𝑒𝑐𝜃=
𝑧= ≈ 103
𝑑𝜃 𝑑𝑡 = 1 25 ∙20∙ ( ) 2
= 4 5 ∙ =.047 𝑟𝑎𝑑/𝑚𝑖𝑛

3. A hot-air balloon rising straight up from a level field is tracked by an inclinometer 500 ft from the lift-off point. At the moment the inclinometer’s angle of elevation is 𝜋 4 radians, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?
𝑑𝜃 𝑑𝑡 =0.14
𝜃= 𝜋 4 y
500 𝜃
𝑡𝑎𝑛𝜃= 𝑦 500
𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 𝑑𝑦 𝑑𝑡
(𝑠𝑒𝑐 𝜋 4 ) 2 ∙0.14= 𝑑𝑦 𝑑𝑡
𝑑𝑦 𝑑𝑡 =2∙0.14∙500=140 𝑓𝑡/𝑚𝑖𝑛

4. A camera on the ground 200 meters away from a hot air balloon records the balloon rising into the sky at a constant rate of 10 m/sec. How fast is the camera’s angle of elevation changing when the balloon is 150 meter in the air?
𝑑𝑦 𝑑𝑡 =10 z
Finding 𝑑𝜃 𝑑𝑡 y 𝜃
200
𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 𝑑𝑦 𝑑𝑡
𝑡𝑎𝑛𝜃= 𝑦 200
Find z before finding 𝑠𝑒𝑐 2 𝜃
𝑧= =250
𝑑𝜃 𝑑𝑡 = ∙10
𝑑𝜃 𝑑𝑡 = 1 20 ∙ 16 25
=.032 𝑟𝑎𝑑/𝑠𝑒𝑐

5. A plane (alt. 4000 ft) is flying west at 700 ft/s
A plane (alt ft) is flying west at 700 ft/s. A searchlight, under its path, tracks it. How fast is the light pivoting when the plane is 5000 ft west?
𝑡𝑎𝑛𝜃= 𝑥 4000
x(t)
5000 ft
𝑡𝑎𝑛𝜃= 𝑥
𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 𝑑𝑥 𝑑𝑡
4000 𝜃
𝑡𝑎𝑛𝜃= = 5 4
𝜃=0.896 𝑟𝑎𝑑
𝑠𝑒𝑐 𝑑𝜃 𝑑𝑡 = ∙700
2.56 𝑑𝜃 𝑑𝑡 =.175
𝑑𝜃 𝑑𝑡 =.068 𝑟𝑎𝑑𝑖𝑎𝑛 𝑓𝑡/𝑠𝑒𝑐

6. A camera on the ground 300 meters away from the launchpad records a hot air balloon rising at a rate of 10 m/sec. How fast is the camera’s angle of elevation changing when the hot air balloon is 400 meters high?
𝟑 𝟐𝟓𝟎 𝒓𝒂𝒅/𝒔𝒆𝒄