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What is the evaluated value of (sin 30° + cos 60°) - tan 45°?

0. sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1. Value = (1/2 + 1/2) - 1 = 1 - 1 = 0.

Mathematics · Class 9-10 · CBSE

Trigonometric Ratios, Standard Angles and Heights & Distances

Learn the 6 trigonometric ratios (sin, cos, tan, cot, sec, cosec), table of standard angle values (0°, 30°, 45°, 60°, 90°), fundamental Pythagorean identities, and real-world angle of elevation & depression height calculations.

1The Six Trigonometric Ratios of an Acute Angle

Trigonometrydeals with the relationships between the sides and angles of a right-angled triangle. For an acute angle θ, the sides are named:Hypotenuse (H)(opposite to 90°),Opposite / Perpendicular (P)(facing angle θ), andAdjacent / Base (B).

Definitions and Reciprocal Relations of the 6 Trigonometric Ratios

Ratio Name	Abbreviation	Side Ratio Definition	Reciprocal Identity
Sine of θ	sin θ	Perpendicular / Hypotenuse (P / H)	sin θ = 1 / cosec θ
Cosine of θ	cos θ	Base / Hypotenuse (B / H)	cos θ = 1 / sec θ
Tangent of θ	tan θ	Perpendicular / Base (P / B)	tan θ = sin θ / cos θ = 1 / cot θ
Cosecant of θ	cosec θ	Hypotenuse / Perpendicular (H / P)	cosec θ = 1 / sin θ
Secant of θ	sec θ	Hypotenuse / Base (H / B)	sec θ = 1 / cos θ
Cotangent of θ	cot θ	Base / Perpendicular (B / P)	cot θ = cos θ / sin θ = 1 / tan θ

Sine of θ
Abbreviation
sin θ
Side Ratio Definition
Perpendicular / Hypotenuse (P / H)
Reciprocal Identity
sin θ = 1 / cosec θ
Cosine of θ
Abbreviation
cos θ
Side Ratio Definition
Base / Hypotenuse (B / H)
Reciprocal Identity
cos θ = 1 / sec θ
Tangent of θ
Abbreviation
tan θ
Side Ratio Definition
Perpendicular / Base (P / B)
Reciprocal Identity
tan θ = sin θ / cos θ = 1 / cot θ
Cosecant of θ
Abbreviation
cosec θ
Side Ratio Definition
Hypotenuse / Perpendicular (H / P)
Reciprocal Identity
cosec θ = 1 / sin θ
Secant of θ
Abbreviation
sec θ
Side Ratio Definition
Hypotenuse / Base (H / B)
Reciprocal Identity
sec θ = 1 / cos θ
Cotangent of θ
Abbreviation
cot θ
Side Ratio Definition
Base / Perpendicular (B / P)
Reciprocal Identity
cot θ = cos θ / sin θ = 1 / tan θ

2Table of Values for Standard Angles (0° to 90°)

Exact Values of Trigonometric Ratios at Standard Angles

Ratio / Angle	0°	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Not Defined (∞)
cosec θ	Not Defined (∞)	2	√2	2/√3	1
sec θ	1	2/√3	√2	2	Not Defined (∞)
cot θ	Not Defined (∞)	√3	1	1/√3	0

sin θ
0°
0
30° (π/6)
1/2
45° (π/4)
1/√2
60° (π/3)
√3/2
90° (π/2)
1
cos θ
0°
1
30° (π/6)
√3/2
45° (π/4)
1/√2
60° (π/3)
1/2
90° (π/2)
0
tan θ
0°
0
30° (π/6)
1/√3
45° (π/4)
1
60° (π/3)
√3
90° (π/2)
Not Defined (∞)
cosec θ
0°
Not Defined (∞)
30° (π/6)
2
45° (π/4)
√2
60° (π/3)
2/√3
90° (π/2)
1
sec θ
0°
1
30° (π/6)
2/√3
45° (π/4)
√2
60° (π/3)
2
90° (π/2)
Not Defined (∞)
cot θ
0°
Not Defined (∞)
30° (π/6)
√3
45° (π/4)
1
60° (π/3)
1/√3
90° (π/2)
0

3Fundamental Pythagorean Trigonometric Identities

Primary Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1 \iff \sin^2\theta = 1 - \cos^2\theta

Secant-Tangent Identity

1 + \tan^2\theta = \sec^2\theta \iff \sec^2\theta - \tan^2\theta = 1

Cosecant-Cotangent Identity

1 + \cot^2\theta = \text{cosec}^2\theta \iff \text{cosec}^2\theta - \cot^2\theta = 1

4Heights and Distances (Applications of Trigonometry)

Angle of Elevation vs Angle of Depression

Angle Type	Observer Line of Sight	Reference Line	Application Example
Angle of Elevation	Observer looks UPWARDS at an elevated object	Angle between horizontal and upward line of sight	Looking up from ground to top of a tower or building
Angle of Depression	Observer looks DOWNWARDS from a height at an object	Angle between horizontal and downward line of sight	Looking down from a lighthouse or cliff at a ship in the sea

Angle of Elevation
Observer Line of Sight
Observer looks UPWARDS at an elevated object
Reference Line
Angle between horizontal and upward line of sight
Application Example
Looking up from ground to top of a tower or building
Angle of Depression
Observer Line of Sight
Observer looks DOWNWARDS from a height at an object
Reference Line
Angle between horizontal and downward line of sight
Application Example
Looking down from a lighthouse or cliff at a ship in the sea

Real-life example · Solved Tower Height Problem

A ladder leaning against a wall makes an angle of 60° with the ground. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder (L):
• cos(60°) = Base / Hypotenuse = 2.5 / L
• 1/2 = 2.5 / L ⇒ L = 2.5 × 2 = 5 metres.

5Exam Corner: Tips, Mistakes & Board Questions

Critical Board Exam Takeaways

Angle of depression from top of tower to object on ground is always mathematically equal to the angle of elevation of the top of the tower from the object (alternate interior angles between horizontal lines).
sin θ and cos θ values can never exceed 1 (since hypotenuse is the longest side of a right triangle).
tan 45° = 1: whenever angle of elevation is 45°, Height = Distance from base!

Exam question · CBSE Class 10 Board

The shadow of a vertical tower on level ground is √3 times the height of the tower. What is the angle of elevation of the sun?

A.30°
B.45°
C.60°
D.90°

Remember

The 6 ratios relate sides to acute angles: ==sin = P/H, cos = B/H, tan = P/B==.
Key identity: == $\sin^2\theta + \cos^2\theta = 1$ and $\sec^2\theta - \tan^2\theta = 1$ ==.
When angle of elevation is $45^\circ$ , height equals the shadow length.
Angle of elevation looking up equals angle of depression looking down.

Check yourself

Question 1
What is the evaluated value of (sin 30° + cos 60°) - tan 45°?